In Chapter 5, you solved systems of linear equations through substitution, addition, subtraction, multiplication, and division. A similar process is employed in this topic, where you will add, subtract, multiply, divide, or substitute functions. The notation used for this looks like the following:

Given two functions $f(x)$ and $g(x)$:

$\begin{array}{clcl}f(x)+g(x)& \text{ is the same as }& (f+g)(x)& \text{ and means the addition of these two functions}\\ f(x)-g(x)& \text{ is the same as }& (f-g)(x)& \text{ and means the subtraction of these two functions}\\ f(x)\cdot g(x)& \text{ is the same as }& (f\cdot g)(x)& \text{ and means the multiplication of these two functions}\\ f(x)÷g(x)& \text{ is the same as }& (f÷g)(x)& \text{ and means the addition of these two functions}\end{array}$

When encountering questions about operations on functions, you will generally be asked to do two things: combine the equations in some described fashion and to substitute some value to replace the variable in the original equation. These are illustrated in the following examples.

Often, you are asked to evaluate operations on functions where you must substitute some given value into the combined functions. Consider the following.

Composite functions are functions that involve substitution of functions, such as $f(x)$ is substituted for the $x$-value in the $g(x)$ function or the reverse. Which goes where is outlined by the way the equation is written:

$\begin{array}{l}(f\circ g)(x)\text{ means that the }g(x)\text{ function is used to replace the }x\text{-values in the }f(x)\text{ function}\\ (g\circ f)(x)\text{ means that the }f(x)\text{ function is used to replace the }x\text{-values in the }g(x)\text{ function}\end{array}$

The more conventional way to write these composite functions is:

$(f\circ g)(x)=f(g(x))\text{ and }(g\circ f)(x)=g(f(x))$

Consider the following examples of composite functions.

Questions

Perform the indicated operations.

1. $g(a)=a^3+5a^2$
   $f(a)=2a+4$
   Find $g(3)+f(3)$
2. $f(x)=-3x^2+3x$
   $g(x)=2x+5$
   Find ${\displaystyle \frac{f(-4)}{g(-4)}}$
3. $g(x)=-4x+1$
   $h(x)=-2x-1$
   Find $g(5)+h(5)$
4. $g(x)=3x+1$
   $f(x)=x^3+3x^2$
   Find $g(2)\cdot f(2)$
5. $g(t)=t-3$
   $h(t)=-3t^3+6t$
   Find $g(1)+h(1)$
6. $g(x)=x^2-2$
   $h(x)=2x+5$
   Find $g(-6)+h(-6)$
7. $h(n)=2n-1$
   $g(n)=3n-5$
   Find ${\displaystyle \frac{h(0)}{g(0)}}$
8. $g(a)=3a-2$
   $h(a)=4a-2$
   Find $(g+h)(10)$
9. $g(a)=3a+3$
   $f(a)=2a-2$
   Find $(g+f)(9)$
10. $g(x)=4x+3$
    $h(x)=x^3-2x^2$
    Find $(g-h)(-1)$
11. $g(x)=x+3$
    $f(x)=-x+4$
    Find $(g-f)(3)$
12. $g(x)=x^2+2$
    $f(x)=2x+5$
    Find $(g-f)(0)$
13. $f(n)=n-5$
    $g(n)=4n+2$
    Find $(f+g)(-8)$
14. $h(t)=t+5$
    $g(t)=3t-5$
    Find $(h\cdot g)(5)$
15. $g(t)=t-4$
    $h(t)=2t$
    Find $(g\cdot h)(3t)$
16. $g(n)=n^2+5$
    $f(n)=3n+5$
    Find ${\displaystyle \frac{g(n)}{f(n)}}$
17. $g(a)=-2a+5$
    $f(a)=3a+5$
    Find $\left({\displaystyle \frac{g}{f}}\right)(a^2)$
18. $h(n)=n^3+4n$
    $g(n)=4n+5$
    Find $h(n)+g(n)$
19. $g(n)=n^2-4n$
    $h(n)=n-5$
    Find $g(n^2)\cdot h(n^2)$
20. $g(n)=n+5$
    $h(n)=2n-5$
    Find $(g\cdot h)(-3n)$

Solve the following composite functions.

21. $f(x)=-4x+1$
    $g(x)=4x+3$
    Find $(f\circ g)(9)$
22. $h(a)=3a+3$
    $g(a)=a+1$
    Find $(h\circ g)(5)$
23. $g(x)=x+4$
    $h(x)=x^2-1$
    Find $(g\circ h)(10)$
24. $f(n)=-4n+2$
    $g(n)=n+4$
    Find $(f\circ g)(9)$
25. $g(x)=2x-4$
    $h(x)=2x^3+4x^2$
    Find $(g\circ h)(3)$
26. $g(x)=x^2-5x$
    $h(x)=4x+4$
    Find $(g\circ h)(x)$
27. $f(a)=-2a+2$
    $g(a)=4a$
    Find $(f\circ g)(a)$
28. $g(x)=4x+4$
    $f(x)=x^3-1$
    Find $(g\circ f)(x)$
29. $g(x)=-x+5$
    $f(x)=2x-3$
    Find $(g\circ f)(x)$
30. $f(t)=4t+3$
    $g(t)=-4t-2$
    Find $(f\circ g)(t)$

Answer Key 11.2