Precalculus Study Guide
Copyright Maplesoft, a division of Waterloo Maple Inc., 2024
Chapter 6: Composition of Functions
Introduction
The function g⁡x=2⁢x+3 defined on all the real numbers takes a real number such as x=2 and "converts" it to (or maps it to) the real number 7. The function f⁡x=5−4⁢x, also defined on all the real numbers, would take a number such as x=7 and map it to −23.
Following the action of one function by the action of another function is called composition of functions, the subject of this chapter.
The notation for the composition of two functions is sometimes given very formally as
f∘g
to represent the abstract idea of the function, and as
(f∘g)⁡x
to represent the value of the composite function at x.
However, the more descriptive notation f⁡g⁡x shows more dramatically which function is substituted into which, for that is really how the rule for the composition is determined. The rule for the "inner" function g⁡x is substituted for x in the "outer" function f⁡x.
In practice, the difficult part of working with composite functions is determining the domain and range of the composition. Only values of x from the domain of g⁡x are candidates for the domain of the composition f⁡g⁡x. However, if some x causes g⁡x to assume a value that is not in the domain of f⁡x, then that value of x cannot be in the domain of the composition f⁡g⁡x.
Moreover, if the rule for the composite function f⁡g⁡x admits a value that would not have been admissible for g⁡x, then that value cannot be in the domain of the composite. An example would be f⁡x=g⁡x = 1x, for which the rule of the composition is
f⁡g⁡x=1⁡1x = x
Looking at just the rule f⁡g⁡x=x, it is tempting to state the domain is all real numbers, but considering that the domain of g⁡x=1x excludes x=0, the domain of the composite function must also exclude x=0.
Chapter Glossary
The following terms in Chapter 6 are linked to the Maple Math Dictionary.
asymptote
bounded
composition
domain
nonnegative
range
real number
Typical Problems
For the functions f⁡x and g⁡x given in each of Problems 6.1 - 6.2,
determine the domain and range of both f⁡x and g⁡x;
obtain the composition f⁡g⁡x, evaluate f⁡g⁡a, draw the graph of the composition, and determine its domain and range;
obtain the composition g⁡f⁡x, evaluate g⁡f⁡a, draw the graph of the composition, and determine its domain and range.
6.1. f⁡x=x+1,g⁡x=xx2+1 , a=1
6.2. f⁡x=xx+1,g⁡x=1x−1,a=2
Maple Initializations
Before accessing the Maple version of the solutions to the typical problems stated above, initialize Maple by pressing the button provided on the right.
Solutions
Problem 6.1
6.1 - Mathematical Solution
6.1 (a) - Mathematical Solution
The rules f⁡x and g⁡x have their domains and ranges specified in Table 6.1.1, where the symbol ℜ denotes the set of real numbers.
Rule
Domain
Range
f⁡x=x+1
{ x∈ℜ, x≥−1 }
{ y∈ℜ, y≥0 }
g⁡x=xx2+1
{ x∈ℜ }
{ y∈ℜ, −12≤y≤12 }
Table 6.1.1 Rules f⁡x and g⁡x, along with domains and ranges
The domain and range for f⁡x can de deduced from the properties of the square-root function, or from the graph of f⁡x in Figure 6.1.1(a).
The domain for g⁡x can be deduced from its rule, but the range can only be approximated from its graph in Figure 6.1.1(b).
Figure 6.1.1(a) Graph of f⁡x=x+1
Figure 6.1.1(b) Graph of g⁡x=xx2+1
The determination of the exact height to which a function such as g⁡x rises is difficult with just the tools of algebra. Numeric computation is merely suggestive, as the values in Table 6.1.2 indicate.
x
0.8
0.9
1.0
1.1
1.2
g⁡x
0.488
0.497
0.500
0.498
0.492
Table 6.1.2 Values of g⁡x=xx2+1
6.1 (b) - Mathematical Solution
The rule for the composition f∘g is
fg⁡x=xx2+1+1
At x=1 we have
f⁡g⁡1=12+1 = 32=32 = 62
A graph of this composition appears in Figure 6.1.2.
The minimum and maximum values of the fraction g⁡x=xx2+1 are −12 and 12, respectively.
Figure 6.1.2 Graph of f⁡g⁡x=xx2+1+1
That implies the minimum and maximum values of f⁡g⁡x will be 1/2 and 3/2, respectively.
Therefore, g⁡x provides to f⁡x only values in the domain of f⁡x, so there are no values in the domain of g⁡x that cannot be passed along to f⁡x. Consequently, Figure 6.1.2 correctly suggests the domain and range of the composition f⁡g⁡x to be respectively, the reals, and the interval 1/2,3/2.
6.1 (c) - Mathematical Solution
The rule for the composition g∘f is
gf⁡x=x+1x+2
At x=1 we have g⁡f⁡1=23.
A graph of this composition appears in Figure 6.1.3.
This figure suggests the domain for the composition g⁡f⁡x is the set of real numbers x, where x>−1, whereas the range is the interval 0,1/2.
Figure 6.1.3 Graph of g⁡f⁡x=x+1x+2
The domain of f⁡x is the set of real numbers x for which x>−1, while the range of f⁡x is the set of nonnegative reals.
If g⁡x is given such a nonnegative real number, it will produce a nonnegative real number no greater than 12.
Consequently, Figure 6.1.3 suggests the correct domain and range for the composition g⁡f⁡x.
6.1 - Maplet Solution
6.1 (a) - Maplet Solution
The rules f⁡x and g⁡x have their domains and ranges specified in Table 6.1.1, reproduced here for convenience. The symbol ℜ denotes the set of real numbers.
The domain and range for f⁡x can de deduced from the properties of the square-root function, or from a graph. The domain for g⁡x can be deduced from its rule, but the range can only be approximated from its graph. Graphs of f⁡x and g⁡x can be obtained with the Composition Tutor .
Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 6.1.4.
This tutor provides a graph of f⁡x,g⁡x, and a choice of the compositions f⁡g⁡x or g⁡f⁡x.
The determination of the exact height to which a function such as g⁡x rises is difficult with just the tools of algebra.
Figure 6.1.4 Thumbnail image of the Composition Tutor
To launch the Composition Tutor, click the following link: Composition Tutor
6.1 (b) - Maplet Solution
The Composition Tutor will provide f⁡g⁡x, the rule for the composition f∘g.
Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 6.1.5.
In addition to graphs of f⁡x,g⁡x, and f⁡g⁡x, the values of these three functions at x=1 are provided.
This tutor will show that the rule for the composition f∘g is
f⁡g⁡x=xx2+1+1 = x2+x+1x2+1
It will also show that at x=1 we have
Figure 6.1.5 Thumbnail image of the Composition Tutor
f⁡g⁡1=12+1 = 32=32 = 62 = 22 3
For this composition, the Composition Tutor also provides a graph similar to the one in Figure 6.1.6, a reproduction of Figure 6.1.2.
The minimum and maximum values of the fraction g⁡x=xx2+1 are −12 and 12, respectively. That implies the minimum and maximum values of f⁡g⁡x will be 1/2 and 3/2, respectively. Therefore, g⁡x provides to f⁡x only values in the domain of f⁡x, so there are no values in the domain of g⁡x that cannot be passed along to f⁡x.
Consequently, Figure 6.1.2 correctly suggests the domain and range of the composition f⁡g⁡x to be respectively, the reals, and the interval 1/2,3/2.
Figure 6.1.6 Graph of fgx=xx2+1+1
6.1 (c) - Maplet Solution
The Composition Tutor will provide g⁡f⁡x, the rule for the composition g∘f. (Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image at the right.) In addition to graphs of f⁡x,g⁡x, and g⁡f⁡x, the values of these three functions at x=1 are provided.
This tutor will show that the rule for the composition g∘f is
g⁡f⁡x=x+1x+2
g⁡f⁡1=23
Figure 6.1.6 Thumbnail image of the Composition Tutor
For this composition, the Composition Tutor also provides a graph similar to the one in Figure 6.1.7, a reproduction of Figure 6.1.3.
The domain of f⁡x is the set of real numbers x for which x>−1, while the range of f⁡x is the set of nonnegative reals. If g⁡x is given such a nonnegative real number, it will produce a nonnegative real number no greater than 12. Consequently, the figure above suggests the correct domain and range for the composition g⁡f⁡x.
Figure 6.1.7 Graph of g⁡f⁡x=x+1x+2
6.1 - Interactive Solution
6.1 (a) -Interactive Solution
Enter the rule for fx.
Context Panel: Plots≻Plot Builder≻−3≤x≤3
The domain and range of f⁡x can be inferred from its graph.
The domain is the set of all real numbers x satisfying x≥−1, while the range is the set of all real numbers y satisfying y≥0.
Enter the rule for gx. Context Panel: Assign to a Name≻g
Context Panel: Plots≻Plot Builder≻−7≤x≤7
The domain and range of g⁡x can be inferred from its graph.
It appears that the domain of g⁡x is the set of all real numbers, but the range is the closed interval −1/2,1/2. The graph suggests the lowest and highest points on the graph of g⁡x are ⁡−1,−12 and ⁡1,12, respectively. This hypothesis is confirmed by the following calculations.
Type g and press the Enter key.
Context Panel: Optimization≻Optimization Assistant Constraints and Bounds≻Edit Add Bound≻x=−2..2 ≻ Add ≻ Done Minimize≻Solve≻Plot Maximize≻Solve≻Plot
6.1 (b) - Interactive Solution
Enter the given functions
Using the template f:=a→y from the Expression palette, define the function f.
Using the template f:=a→y from the Expression palette, define the function g.
Obtain and graph the composition fgx
Type fgx and press the Enter key.
Context Panel: Plots≻Plot Builder −10≤x≤10 Options≻0≤y≤1.3
Evaluations at x=a=1
Context Panel: Evaluate at a Point≻1
Type g1
Context Panel: Assign to a Name≻ga
Type fga and press the Enter key.
Domain and Range of the composition fgx
Figure 6.1.2 suggests the domain for the composition f⁡g⁡x is the set of real numbers, whereas the range is a bounded set localized about y=1. The best tools for determining the maximum and minimum values of a function are developed in the calculus. Here, however, we can do the following.
The minimum and maximum values of the fraction g⁡x=xx2+1 are −1/2 and 1/2, respectively. That implies the minimum and maximum values of f⁡g⁡x will be 1/2 and 3/2, respectively.
The extreme values of gx can be obtained numerically as follows.
6.1 (c) - Interactive Solution
Obtain and graph the composition gfx
Type gfx and press the Enter key.
Context Panel: Plots≻Plot Builder −4≤x≤10 Options≻−110≤y≤35 Options≻Number of Points≻1000
Type f1
Context Panel: Assign to a Name≻fa
Type gfa and press the Enter key.
Domain and Range of the composition gfx
Figure 6.1.3 suggests the domain for the composition g⁡f⁡x is the set of real numbers x, where x≥−1, whereas the range is the interval 0,1/2.
The domain of f⁡x is the set of real numbers x for which x≥−1, while the range of f⁡x is the set of nonnegative reals.
If g⁡x is given such a nonnegative real number, it will produce a nonnegative real number no greater than 1/2.
The maximum value of the composition gfx can be determined numerically via the Optimization Assistant.
Context Panel: Optimization≻Optimization Assistant Maximize≻Solve≻Plot
6.1 - Programmatic Solution
6.1 (a) - Programmatic Solution
Enter the functions f and g.
f≔x→x+1; g≔x→xx2+1
Graph f as per Figure 6.1.1(a).
plotf,−3..3
From this graph, or from the rule for f, infer that the domain of f is the interval −1,∞, and that the range is the interval 0,∞.
Graph g as per Figure 6.1.1(b).
plotg,−7..7
That the domain of g is the set of all real numbers can be inferred from Figure 6.1.1(b). It is more difficult to determine that the range is the interval −1/2,1/2.
Obtain the minimum and maximum for g. The minimize and maximize commands provide the extreme values as well as the x-coordinate where the extreme occurs.
minimizegx,location2;maximizegx,location2
6.1 (b) - Programmatic Solution
Obtain the composition fgx.
fg≔fgx
Obtain fg1.
evalfg,x=1
Obtain g1.
g1
Obtain f12=fg1.
f12
Graph fgx, thereby obtaining Figure 6.1.2.
plotfgx,x=−10..10,y=0..1.3
Figure 6.1.2 suggests the domain for the composition f⁡g⁡x is the set of real numbers, whereas the range is a bounded set localized about y=1. The best tools for determining the maximum and minimum values of a function are developed in the calculus. Here, however, we reason as follows.
The minimum and maximum values of the fraction g⁡x=xx2+1 are −12 and 12, respectively. That implies the minimum and maximum values of f⁡g⁡x will be 1/2 and 3/2, respectively.
Obtain the minimum and maximum for fgx. The minimize and maximize commands provide the extreme values as well as the x-coordinate where the extreme occurs.
minimizefgx,location2;maximizefgx,location2
6.1 (c) - Programmatic Solution
Obtain the composition gfx.
gf≔gfx
Compute gf1.
evalgf,x=1
Obtain f1.
f1
Obtain g2=gf1.
g2
Graph gfx, thereby obtaining Figure 6.1.3.
plotgf,x=−4..10,y=−110..35,numpoints=3000
Obtain the minimum and maximum for fgx. The maximize command provides the extreme values as well as the x-coordinate where the extreme occurs.
maximizegf,x=−12..3, location2
Problem 6.2
6.2 - Mathematical Solution
6.2 (a) - Mathematical Solution
The rules f⁡x and g⁡x have their domains and ranges specified in Table 6.2.1, where the symbol ℜ denotes the set of real numbers.
f⁡x=xx+1
{ x∈ℜ, x≠−1 }
{ y∈ℜ, y≠1 }
g⁡x=1x−1
{ x∈ℜ,x≠1 }
{ y∈ℜ, y≠0 }
Table 6.2.1 Rules f⁡x and g⁡x, along with domains and ranges
The domain and range for f⁡x can be deduced from the properties of the fraction xx+1, or from the graph of f⁡x in Figure 6.2.1(a).
The vertical asymptote whose equation is x=−1, and the horizontal asymptote y=1 determine the domain and range, respectively.
Consequently, the domain consists of all real numbers except the number −1, whereas the range consists of all real numbers except the number 1.
Figure 6.2.1(a) Graph of f⁡x=xx+1
The domain and range for g⁡x can be deduced from the properties of the fraction 1x−1, or from the graph of g⁡x in Figure 6.2.1(b).
The vertical asymptote whose equation is x=1, and the horizontal asymptote y=0 determine the domain and range, respectively.
Consequently, the domain consists of all real numbers except the number 1, whereas the range consists of all real numbers except the number 0.
Figure 6.2.1(b) Graph of g⁡x=1x−1
6.2 (b) - Mathematical Solution
f⁡g⁡x=⁡1x−11x−1+1 = 11+x−1=1x
At x=2 we have f⁡g⁡2=12.
A graph of this composition appears in Figure 6.2.2.
Figure 6.2.2 seems to suggest the domain for the composition f⁡g⁡x is the set of real numbers x satisfying x≠0, and the range is the set of real numbers y satisfying y≠0. However, this is naive. The analysis must be more sophisticated.
Figure 6.2.2 Graph of fg⁡x=1x
The "inside" function g⁡x=1x−1 cannot be given x=1. Hence, the domain of the composition f⁡g⁡x cannot contain x=1. Moreover, the function f⁡x=xx+1 cannot be given x=−1. We must determine which value, if any, in the domain of g⁡x, delivers to f⁡x the value −1. Thus, we must solve the equation
g⁡x=1x−1 = −1
obtaining
1=1−x
from which it follows that x=0. Thus, the two real numbers x=0 and x=1, cannot be in the domain of the composition f⁡g⁡x. Consequently, if x=1 cannot be in the domain of the rule f⁡g⁡x=1x, then y=1 will not be in the range of the composition. Hence, the range of f⁡g⁡x is the set of real numbers y, except for the two values y=0, and y=1.
6.2 (c) - Mathematical Solution
g⁡f⁡x=1⁡xx+1−1 = x+1x−x−1=−x+1
At x=2 we have g⁡f⁡2=−3.
A graph of this composition appears in Figure 6.2.3.
It might appear from Figure 6.2.3 that both the domain and range for the composition f⁡g⁡x=−x+1 would be the set of all real numbers. However, that is naive. A more sophisticated analysis is required.
Figure 6.2.3 Graph of gfx= −x+1
The domain of f⁡x=xx+1 is the set of real numbers x for which x≠−1, while the range of f⁡x is the set of reals y for which y≠1. Therefore, f⁡x will never give g⁡x the value 1, which is the only input value it must not receive, so the only real x not in the domain of the composition is x=−1. Now, if x=−1 will never be given to the rule g⁡f⁡x=−x+1, then the range of the composition will never contain y=0.
The domain of the composition g⁡f⁡x is the set of real numbers x for which x≠−1, whereas the range is the set of real numbers y for which y≠0.
6.2 - Maplet Solution
6.2 (a) - Maplet Solution
The rules f⁡x and g⁡x have their domains and ranges specified in Table 6.2.1, reproduced here for convenience. The symbol ℜ denotes the set of real numbers.
The domain and range for f⁡x can de deduced from the properties of the fraction xx+1, or from a graph. The vertical asymptote whose equation is x=−1, and the horizontal asymptote y=1 determine the domain and range, respectively.
The domain and range for g⁡x can be deduced from the properties of the fraction 1x−1, or from a graph. The vertical asymptote whose equation is x=1, and the horizontal asymptote y=0 determine the domain and range, respectively.
Graphs of f⁡x and g⁡x can be obtained with the Composition Tutor .
Figure 6.2.4 Thumbnail image of the Composition Tutor
Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 6.2.4.
6.2 (b) - Maplet Solution
Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 6.2.5.
In addition to graphs of f⁡x,g⁡x, and f⁡g⁡x, the values of these three functions at x=2 are provided.
Figure 6.2.5 Thumbnail image of the Composition Tutor
It will also show that at x=2 we have f⁡g⁡2=12.
For this composition, the Composition Tutor also provides a graph similar to the one in Figure 6.2.6, a reproduction of Figure 6.2.2. This graph seems to suggest the domain for the rule f⁡g⁡xis the set of real numbers x satisfying x≠0, and the range is the set of real numbers y satisfying y≠0. However, this is naive. The analysis must be more sophisticated.
Figure 6.2.6 Graph of fgx=1x
from which it follows that x=0. Thus, the two real numbers 0 and 1 cannot be in the domain of the composition f⁡g⁡x. Consequently, if x=1 cannot be in the domain of the rule f⁡g⁡x=1x, then y=1 will not be in the range of the composition. Hence, the range of f⁡g⁡x is the set of real numbers y, except for the two values y=0, and y=1.
6.2 (c) - Maplet Solution
The Composition Tutor will provide g⁡f⁡x, the rule for the composition g∘f.
Clicking this link will launch the tutor with the solution embedded as shown in the thumbnail image in Figure 6.2.7.
In addition to graphs of f⁡x,g⁡x, and g⁡f⁡x, the values of these three functions at x=2 are provided.
It will also show that at x=2 we have g⁡f⁡2=−3.
Figure 6.2.7 Thumbnail image of the Composition Tutor
For this composition, the Composition Tutor also provides a graph similar to the one in Figure 6.2.8, a reproduction of Figure 6.2.3.
It might appear from Figure 6.2.8 (same as Figure 6.2.3) that both the domain and range for the composition gf⁡x=−x+1 would be the set of all real numbers However, that is naive. A more sophisticated analysis is required.
Figure 6.2.8 Graph of gf⁡x= −x+1
6.2 - Interactive Solution
6.2 (a) - Interactive Solution
Context Panel: Plots≻Plot Builder −3≤x≤3 Options≻−10≤y≤10 Options≻Find Discontinuities
The domain is the set of all real numbers x satisfying x≠−1, while the range is the set of all real numbers y satisfying y≠1.
Context Panel: Plots≻Plot Builder≻ −7≤x≤7 Options≻−5≤y≤5 Options≻Find Discontinuities
The domain of g⁡x is the set of all real numbers x for which x≠1, whereas the range is the set of all real numbers y for which y≠0.
6.2 (b) - Interactive Solution
Context Panel: Plots≻Plot Builder −5≤x≤5 Options≻−5≤y≤5
Evaluations at x=a=2
Context Panel: Evaluate at a Point≻2
Type g2
The "inside" function g⁡x=1x−1 cannot be given x=1. Hence, the domain of the composition f⁡g⁡x cannot contain x=1. Moreover, the function f⁡x=xx+1 cannot be given x=−1. We must determine which value, if any, in the domain of g⁡x, delivers to f⁡x the value −1. Thus, we must solve the equation g⁡x=−1:
Enter the equation gx=−1
Context Panel: Solve≻Solve
Thus, the two real numbers x=0 and x=1, cannot be in the domain of the composition f⁡g⁡x. Consequently, if x=1 cannot be in the domain of the rule f⁡g⁡x=1x, then y=1 will not be in the range of the composition. Hence, the range of f⁡g⁡x is the set of real numbers y, except for the two values y=0, and y=1.
6.2 (c) - Interactive Solution
Context Panel: Plots≻Plot Builder −4≤x≤4
Type f2
Figure 6.2.3, suggests the domain for the composition g⁡f⁡x is the set of all real numbers x and the range is the set of all real numbers y. However, this is naive. The analysis must be more sophisticated.
The domain of f⁡x=xx+1 is the set of real numbers x for which x≠−1, while the range of f⁡x is the set of real numbers y for which y≠1. Therefore, f⁡x will never give g⁡x the value 1, which is the only input value it must not receive, so the only real x not in the domain of the composition is x=−1. Now, if x=−1 will never be given to the rule g⁡f⁡x=−x+1, then the range of the composition will never contain y=0.
6.2 - Programmatic Solution
6.2 (a) - Programmatic Solution
f≔x→xx+1; g≔x→1x−1
Graph f as per Figure 6.2.1(a).
plotfx,x=−3..3,y=−10..10,discont=true
The domain of f is the set of all real numbers x satisfying x≠−1, while the range is the set of all real numbers y satisfying y≠1.
Graph f as per Figure 6.2.1(b).
plotgx,x=−2..5,y=−5..5,discont=true
The domain and range of g⁡x can be inferred from its graph in Figure 6.2.1(b).
6.2 (b) - Programmatic Solution
fg≔simplifyfgx
Obtain fg2.
evalfg,x=2
Obtain g2.
Obtain f1=fg2.
Graph fgx, thereby obtaining Figure 6.2.2.
plotfgx,x=−5..5,y=−5..5
A graph of the rule f⁡g⁡x=1x appears in Figure 6.2.2, a figure that seems to suggest the domain for the composition f⁡g⁡x is the set of real numbers x satisfying x≠0, and the range is the set of real numbers y satisfying y≠0. However, this is naive. The analysis must be more sophisticated.
The "inside" function g⁡x=1x−1 cannot be given x=1. Hence, the domain of the composition f⁡g⁡x cannot contain x=1. Moreover, the function f⁡x=xx+1 cannot be given x=−1. We must determine which value, if any, in the domain of g⁡x, delivers to f⁡x the value −1. Thus, we must solve the equation g⁡x=−1, obtaining x=0, as seen from the following calculation.
Solve the equation gx= −1.
solvegx=−1,x
6.2 (c) - Programmatic Solution
gf≔simplifygfx
Compute gf2.
evalgf,x=2
Obtain f2.
f2
Obtain g2/3=gf2.
g2/3
plotgf,x=−4..4
A graph of the rule g⁡f⁡x=−x+1 appears at the right, in Figure 6.2.3, a figure that suggests the domain for the composition g⁡f⁡x is the set of all real numbers x and the range is the set of all real numbers y. However, this is naive. The analysis must be more sophisticated.
Exercises - Chapter 6
For the functions f⁡x and g⁡x given in each of Exercises 6.1 - 6.10,
(a) determine the domain and range of both f⁡x and g⁡x;
(b) obtain the composition f⁡g⁡x, evaluate f⁡g⁡a, draw the graph of the composition, and determine its domain and range;
(c) obtain the composition g⁡f⁡x, evaluate g⁡f⁡a, draw the graph of the composition, and determine its domain and range.
The domains for the functions f⁡x and g⁡x are the largest set of real numbers for which the given rules are defined. Where necessary, estimate bounds on any domain or range graphically.
6.1. f⁡x=3⁢x−4,g⁡x=7−5⁢x,a=3
6.2. f⁡x=4⁢x+5,g⁡x=1−2⁢x,a=−2
6.3. f⁡x=5⁢x+7,g⁡x=3⁢x2+2⁢x−1,a=2
6.4.