Chapter 11: Functions

# 11.2 Operations on Functions

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 $g(x)$:

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

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.

Example 11.2.1

Perform the following operations on $f(x) = 2x^2 - 4$ and $g(x) = x^2 + 4x - 2$.

1. $f(x) + g(x)$Addition yields $2x^2 - 4 + x^2 + 4x - 2$, which simplifies to $3x^2 + 4x - 6$.
2. $f(x) - g(x)$Subtraction yields $2x^2-4-(x^2+4x-2)$, which simplifies to $x^2-4x-2$.
3. $f(x)\cdot g(x)$Multiplication yields $(2x^2-4)(x^2+4x-2)$, which simplifies to $2x^4+8x^3-4x^2-16x+8$.
4. $f(x)\div g(x)$Division yields $(2x^2-4)\div (x^2+4x-2)$, which cannot be reduced any further.

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

Example 11.2.2

Perform the following operations on $f(x) = x^2 - 3$ and $g(x) = 2x^2 + 3x$ and evaluate for the given values.

1. $f(2) + g(2)$
$[x^2-3]+[2x^2+3x]$
$[(2)^2-3]+[2(2)^2+3(2)]$
$4-3+8+6=15$
$f(2)+g(2)=15$
2. $f(1) - g(3)$
$[x^2-3]-[2x^2+3x]$
$[(1)^2-3]-[2(3)^2+3(3)]$
$[1-3]-[18+9]=-29$
$f(1)-g(3)=-29$
3. $f(0)\cdot g(2)$
$[x^2-3]\cdot [2x^2+3x]$
$[0^2-3]\cdot [2(2)^2+3(2)]$
$[-3]\cdot [8+6]=-42$
$f(0)\cdot g(2)=-42$
4. $f(2)\div g(0)$
$[x^2-3]\div [2x^2+3x]$
$[2^2-3]\div [2(0)^2+3(0)]$
$[1]\div [0]=\text{ undefined}$

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.

Example 11.2.3

Given the functions $f(x) = 3x - 5$ and $g(x) = x^2 + 2$, evaluate for:

1. $(f\circ g)(2)$$\begin{array}{rrl} (f\circ g)(x)&=&f(g(x)) \\ f(g(x))&=&3(x^2+2)-5 \\ f(g(2))&=&3(2^2+2)-5 \\ f(g(2))&=&3(6)-5=13 \end{array}$
2. $(g\circ f)(-1)$$\begin{array}{rrl} (g\circ f)(x)&=&g(f(x)) \\ g(f(x))&=&[3x-5]^2+2 \\ g(f(-1))&=&[3(-1)-5]^2+2 \\ g(f(-1))&=&[-8]^2+2 \\ g(f(-1))&=&66 \end{array}$

# 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 $\dfrac{f(-4)}{g(-4)}$
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 $\dfrac{h(0)}{g(0)}$
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 $\dfrac{g(n)}{f(n)}$
17. $g(a) = -2a + 5$
$f(a) = 3a + 5$
Find $\left(\dfrac{g}{f}\right)(a^2)$
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.

1. $f(x) = -4x + 1$
$g(x) = 4x + 3$
Find $(f\circ g)(9)$
2. $h(a) = 3a + 3$
$g(a) = a + 1$
Find $(h \circ g)(5)$
3. $g(x) = x + 4$
$h(x) = x^2 - 1$
Find $(g \circ h)(10)$
4. $f(n) = -4n + 2$
$g(n) = n + 4$
Find $(f \circ g)(9)$
5. $g(x) = 2x - 4$
$h(x) = 2x^3 + 4x^2$
Find $(g \circ h)(3)$
6. $g(x) = x^2 - 5x$
$h(x) = 4x + 4$
Find $(g \circ h)(x)$
7. $f(a) = -2a + 2$
$g(a) = 4a$
Find $(f \circ g)(a)$
8. $g(x) = 4x + 4$
$f(x) = x^3 - 1$
Find $(g \circ f)(x)$
9. $g(x) = -x + 5$
$f(x) = 2x - 3$
Find $(g \circ f)(x)$
10. $f(t) = 4t + 3$
$g(t) = -4t - 2$
Find $(f \circ g)(t)$