Operations on Functions

Section 10.2 Operations on Functions

Objective: Combine functions using sum, difference, product, quotient and composition of functions.

Several functions can work together in one larger function. There are $5$ common operations that can be performed on functions. The four basic operations on functions are adding, subtracting, multiplying, and dividing. The notation for these functions is as follows.

\begin{align*} \text{Addition} \amp \quad (f+g)(x)=f(x)+g(x) \\ \text{Subtraction} \amp\quad (f-g)(x)=f(x)-g(x) \\ \text{Multiplication} \amp\quad (f \cdot g)(x)=f(x)g(x) \\ \text{Division} \amp\quad \left(\frac{f}{g}\right)(x)= \frac{f(x)}{g(x)} \end{align*}

When we do one of these four basic operations we can simply evaluate the two functions at the value and then do the operation with both solutions

Example 10.2.1.

\begin{align*} f(x)=x^2-x-2 \amp \quad \\ g(x)=x+1 \amp\quad \text{Evaluate $f$ and $g$ at } -3 \\ \text{find } (f+g)(-3) \amp\quad \\ \\ f(-3)=(-3)^2 –(-3)-2 \amp\quad \text{Evaluate $f$ at } -3 \\ f(-3)=9+3-2 \amp\quad \\ f(-3)=10 \amp\quad \\ \\ g(-3)=(-3)+1 \amp\quad \text{ Evaluate $g$ at } -3 \\ g(-3)=-2 \amp\quad \\ \\ f(-3)+g(-3) \amp\quad \text{Add the two functions together} \\ (10)+(-2) \amp\quad \text{Add} \\ 8 \amp\quad \text{Our Solution} \end{align*}

The process is the same regardless of the operation being performed.

Example 10.2.2.

\begin{align*} h(x)=2x-4 \amp \quad \\ k(x)=-3x+1 \amp\quad \text{Evaluate $h$ and $k$ at} 5 \\ \text{find } (h\cdot k)(5) \amp\quad \\ \\ h(5)=2(5)-4 \amp\quad \text{Evaluate $h$ at } 5 \\ h(5)=10-4 \amp\quad \\ h(5)=6 \amp\quad \\ \\ k(5)=-3(5)+1 \amp\quad \text{ Evaluate $k$ at } 5 \\ k(5)=-15+1 \amp\quad \\ k(5)=-14 \amp\quad \\ \\ h(5)k(5) \amp\quad \text{Multiply the two results together} \\ (6)(-14) \amp\quad \text{ Multiply } \\ -84 \amp\quad \text{Our Solution} \end{align*}

Often as we add, subtract, multiply, or divide functions, we do so in a way that keeps the variable. If there is no number to plug into the equations we will simply use each equation, in parenthesis, and simplify the expression.

Example 10.2.3.

\begin{align*} f(x)=2x-4 \amp \quad \\ g(x)=x^2-x+5 \amp\quad \text{Write subtraction problem of functions} \\ \text{Find } (f-g)(x) \amp\quad \\ \\ f(x)-g(x) \amp\quad \text{Replace $f(x)$ with $(2x-3)$ and $g(x)$ with } (x^2-x+5) \\ (2x-4)-(x^2-x+5) \amp\quad \text{Distribute the negative} \\ 2x-4-x^2+x-5 \amp\quad \text{Combine like terms} \\ -x^2+3x-9 \amp\quad \text{Our Solution} \end{align*}

The parentheses are very important when we are replacing $f(x)$ and $g(x)$ with a variable. In the previous example we needed the parenthesis to know to distribute the negative.

Example 10.2.4.

\begin{align*} f(x)=x^2-4x-5 \amp \quad \\ g(x)=x-5 \amp\quad \text{Write division problem of functions} \\ \text{Find } \left(\frac{f}{g}\right)(x) \amp\quad \\ \\ \frac{f(x)}{g(x)} \amp\quad \text{Replace $f(x)$ with $(x^2-4x-5)$ and $g(x)$ with } (x-5) \\ \frac{(x^2-4x-5)}{(x-5)}\amp\quad \text{To simplify the fraction we must first factor } \\ \frac{(x-5)}{(x+1)} \amp\quad \text{Divide out common factor of } x-5 \\ x+1 \amp\quad \text{Our Solution} \end{align*}

Just as we could substitute an expression into evaluating functions, we can substitute an expression into the operations on functions.

Example 10.2.5.

\begin{align*} f(x)=2x-1 \amp \quad \\ g(x)=x+4 \amp\quad \text{Write as a sum of functions} \\ \text{Find } (f+g)(x^2) \amp\quad \\ \\ f(x^2)+g(x^2) \amp\quad \text{Replace $x$ in $f(x)$ and $g(x)$ with } (x-5) \\ [2(x^2)-1]+[(x^2)+4]\amp\quad \text{To simplify the fraction we must first factor } \\ 2x^2-1+x^2+4 \amp\quad \text{Distribute the $+$ does not change the problem} \\ 3x^3+3 \amp\quad \text{Our Solution} \end{align*}

Example 10.2.6.

\begin{align*} f(x)=2x-1 \amp \quad \\ g(x)=x+4 \amp\quad \text{Write as a product of functions} \\ \text{Find } (f\cdot g)(3x) \amp\quad \\ \\ f(3x)g(3x) \amp\quad \text{Replace $x$ in $f(x)$ and $g(x)$ with } (3x) \\ [2(3x)-1]+[(3x)+4]\amp\quad \text{Multiply our }2(3x) \\ (6x-1)(3x+4)\amp\quad \text{FOIL} \\ 18x^2+24x-3x-4 \amp\quad \text{Combine like terms } \\ 18x^2+21x-4 \amp\quad \text{Our Solution} \end{align*}

The fifth operation of functions is called composition of functions. A composition of functions is a function inside of a function. The notation used for composition of functions is:

\begin{equation*} \mathbf{ (f\circ g)(x)=f(g(x))} \end{equation*}

To calculate a composition of function we will evaluate the inner function and substitute the answer into the outer function. This is shown in the following example.

Example 10.2.7.

\begin{align*} a(x)=x^2-2x+1 \amp \quad \\ b(x)=x-5 \amp\quad \text{Rewrite as a function in function} \\ \text{Find } (a\circ b)(3) \amp \quad \\ \\ a(b(3)) \amp\quad \text{Evaluate the inner function in function} \\ b(3)=(3)-5=-2 \amp \quad \text{This solution is put into } a, a(-2) \\ a(-2)=(-2)^2-2(-2)+1 \amp\quad \text{Evaluate} \\ a(-2)=4+4+1 \amp\quad \text{Add} \\ a(-2)=9 \amp\quad \text{Our Solution} \end{align*}

We can also evaluate a composition of functions at a variable. In these problems we will take the inside function and substitute into the outside function.

Example 10.2.8.

\begin{align*} f(x)=x^2-x \amp \quad \\ g(x)=x+3 \amp\quad \text{Rewrite as a function in function} \\ \text{Find } (f\circ g)(x) \amp \quad \\ \\ f(g(x)) \amp\quad \text{Replace $g(x)$ with } x+3 \\ f(x)=(x+3) \amp \quad \text{Replace the variable in $f$ with } (x+3) \\ (x+3)^2-(x+3) \amp\quad \text{Evaluate exponent} \\ (x^2+6x+9)-(x-3) \amp\quad \text{Distribute negative} \\ x^2+6x+9-x-3 \amp\quad \text{Combine like terms } \\ x^2+5x+6 \amp\quad \text{Our Solution} \end{align*}

It is important to note that very rarely is $(f \circ g)(x)$ the same as $(g \circ f)(x)$ as the following example will show, using the same equations, but compositing them in the opposite direction.

Example 10.2.9.

\begin{align*} f(x)=x^2-x \amp \quad \\ g(x)=x+3 \amp\quad \text{Rewrite as a function in function} \\ \text{Find } (g\circ f)(x) \amp \quad \\ \\ g(f(x)) \amp\quad \text{Replace $f(x)$ with } x^2-x \\ g(x^2-x) \amp \quad \text{Replace the variable in $g$ with } (x^2-x) \\ (x^2-x)+3 \amp\quad \text{Here the parenthesis don’t change the expression} \\ x^2-x+3 \amp\quad \text{Our Solution} \end{align*}

World View Note: The term “function” came from Gottfried Wihelm Leibniz, a German mathematician from the late $17$th century.

Exercises Exercises – Operations on Functions

Exercise Group.

Perform the indicated operations.

1.

$g(a)=a^3 +5a^2$

$\quad f(a)=2a+4 $

$\quad \text{Find } g(3)+f(3) $

2.

$f(x)=-3x^3 +3x$

$\quad g(x)=2x+5 $

$\quad \text{Find } f(-4) \div g(-4) $

3.

$g(a)=3a +3$

$\quad f(a)=2a-2 $

$\quad \text{Find } (g + f)(9) $

4.

$g(x)=4x+3$

$\quad h(x)=x^3 + 3x^2 $

$\quad \text{Find } (g-h)(-1)$

5.

$g(x)=x+3$

$\quad f(x)=-x+4 $

$\quad \text{Find } (g -f)(3) $

6.

$g(x)=-4x+1$

$\quad h(x)=-2x-1 $

$\quad \text{Find } g(5)+h(5) $

7.

$g(x)=x^2 +2$

$\quad f(x)=2x+5 $

$\quad \text{Find } (g-f)(0) $

8.

$g(a)=3x+1$

$\quad f(a)=x^3+3x^2 $

$\quad \text{Find } g(2) \cdot f(2) $

9.

$g(t)=t-3$

$\quad h(t)=-3T^3 +6t $

$\quad \text{Find } g(1)+h(1) $

10.

$f(n)=n-5$

$\quad g(n)=4n+2 $

$\quad \text{Find } (f+g)(-8)$

11.

$h(t)=t+5$

$\quad g(t)=3t-5 $

$\quad \text{Find } (h \cdot g)(5) $

12.

$g(a)=3a -2$

$\quad h(a)=4a+2 $

$\quad \text{Find } (g+h)(-10) $

13.

$g(n)=2n-1$

$g(n)=3n-5 $

$\quad \text{Find } h(0) \div g(0) $

14.

$g(x)=x^2 -2$

$\quad h(x)=2x+5 $

$\quad \text{Find } g(-6)+h(-6) $

15.

$f(a)=-2a - 4$

$\quad g(a)=a^2 + 3 $

$\quad \text{Find } \left(\frac{f}{g}\right)(7) $

16.

$g(n) = n^-3$

$\quad h(n)=2n-3 $

$\quad \text{Find } (g – h)(n) $

17.

$g(x)=- x^3 - 2$

$\quad h(x)= 4x $

$\quad \text{Find } (g – h)(x) $

18.

$g(x)=2x - 3$

$\quad h(x)=x^3 - 2x^2 + 2x $

$\quad \text{Find } (g – h)(x) $

19.

$f(x)= - 3x + 2$

$\quad g(x)= x^2 + 5x $

$\quad \text{Find } (f - g)(x) $

20.

$g(t)=t - 4$

$\quad h(a)= 2t$

$\quad \text{Find } (g \cdot h)(t) $

21.

$g(x)=4x + 5$

$\quad h(x)= x^2 + 5x $

$\quad \text{Find } g(x) \cdot h(x) $

22.

$g(t)= - 2t^2 – 5t $

$\quad h(t)=t + 5 $

$\quad \text{Find } g(x) \cdot h(x) $

23.

$f(x)=x^2 - 5x$

$\quad g(x)=x+5 $

$\quad \text{Find } ( f + g)( x ) $

24.

$f(x)=4x -4$

$\quad g(x)=3x^2 - 5 $

$\quad \text{Find } ( f + g)( x ) $

25.

$g(n)=n^2 + 5$

$\quad f(n)= 3n + 5 $

$\quad \text{Find } g(n) \div f(n) $

26.

$f(x)=2x + 4$

$\quad g(x)= 4x - 5 $

$\quad \text{Find } f(x) – g(x) $

27.

$g(a)=-2a +5$

$\quad f(a)=3a + 5 $

$\quad \text{Find } \left(\frac{g}{f}\right)(a) $

28.

$g(t)=t^3 + 3t^2 $

$\quad h(t)=3t - 5 $

$\quad \text{Find } g(3)+f(3) $

29.

$h(n) = n^3 + 4n$

$\quad g(n)=4n +5 $

$\quad \text{Find } h(n) + g(n) $

30.

$f(x)=4x +2$

$\quad g(x)=x^2 + 2x $

$\quad \text{Find } f(x) \div g(x) $

31.

$g(n)=n^2 - 4n$

$\quad h(n)= n - 5 $

$\quad \text{Find } g(n^2) \cdot h(n^2) $

32.

$g(n)=n + 5$

$\quad f(a)=2n - 5 $

$\quad \text{Find } (g\cdot h)(-3n) $

33.

$f(x)=2x$

$\quad g(x)= -3x -1 $

$\quad \text{Find } (f+g)( -4 -x) $

34.

$g(a)=-2a$

$\quad h(a)=3a $

$\quad \text{Find } g(4a) \ frac h(4a) $

35.

$f(t)=t^2 + 4t$

$\quad g(t)=4t + 2$

$\quad \text{Find } f(t^2) + g(t^2) $

36.

$h(n)=3n - 2$

$\quad g(n)=- 3n^2 – 4n $

$\quad \text{Find } g(3)+f(3) $

37.

$g(a)=a^3 + 2a$

$\quad h(a)=3a+4 $

$\quad \text{Find } (g \div h) (3) $

38.

$g(x)=- 4x + 2 $

$\quad h(x)=x^2 - 5 $

$\quad \text{Find } g(x^2)+h(x^2) $

39.

$f(n) = -3n^2 + 1$

$\quad g(n)= 2n + 1 $

$\quad \text{Find } (f – g)(n \ frac 3) $

40.

$f(n)=3n + 4 $

$\quad g(n)=n^3 – 5n $

$\quad \text{Find } f(n \div 2) - g(n \div 2) $

41.

$f(x)=- 4x + 1$

$\quad g(x)= 4x + 3 $

$\quad \text{Find } ( f \circ g) (9)$

42.

$g(x)=x - 1$

$\quad \text{Find } ( g \circ g) (7)$

43.

$h(a)=3a + 3$

$\quad g(a)=a + 1 $

$\quad \text{Find } (h \circ g) (5) $

44.

$g(t)=t + 3$

$\quad h(t)=2t -5 $

$\quad \text{Find } (g \circ h) (3) $

45.

$g(x)=x + 4 $

$\quad h(x)=x^2 -1 $

$\quad \text{Find } (g \circ h ) (10) $

46.

$f(a)=2a-4$

$\quad g(a)=a^2+2a $

$\quad \text{Find } (f \circ g)(-4) $

47.

$f(n)=-4n+2$

$\quad g(n)=n+4 $

$\quad \text{Find } (f \circ g)(9) $

48.

$g(x)=3x+4$

$\quad h(x)=x^3+3x$

$\quad \text{Find } (g \circ h)(3) $

49.

$g(x)=2x-4$

$\quad h(x)= 2x^3+4x^2$

$\quad \text{Find } (g \circ h)(3) $

50.

$g(a)=a^2+3$

$\quad \text{Find } (g \circ g)(-3) $

51.

$g(x)=x^2-5x$

$\quad h(x)=4x+4 $

$\quad \text{Find } (g \circ h)(x) $

52.

$g(a)=2a+4 $

$\quad h(a)=-4a+5 $

$\quad \text{Find } (g \circ h)(a) $

53.

$f(a)=-2a+2$

$\quad g(a)=4a $

$\quad \text{Find } (f \circ g)(a) $

54.

$g(t)=-t-4$

$\quad \text{Find } (g \circ g)(t) $

55.

$g(x)=4x+4$

$\quad f(x)= x^3-1 $

$\quad \text{Find } (g \circ f)(x) $

56.

$f(n)=-2n^2-4n$

$\quad g(n)=n+2 $

$\quad \text{Find } (f \circ g)(n) $

57.

$g(t)=-x+5$

$\quad f(t)=2x-3 $

$\quad \text{Find } (g \circ f)(x) $

58.

$g(t)=t^3-t$

$\quad f(t)=3t-4 $

$\quad \text{Find } (g \circ f)(t) $

59.

$f(t)=4t+3$

$\quad g(t)=-4t-2 $

$\quad \text{Find } (f \circ g)(t) $

60.

$f(x)=3x- 4$

$\quad g(x)=x^3+2x^2 $

$\quad \text{Find } (f \circ g)(x) $

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