Section 10.2 Operations on 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.
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.
The process is the same regardless of the operation being performed.
Example 10.2.2.
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.
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.
Just as we could substitute an expression into evaluating functions, we can substitute an expression into the operations on functions.
Example 10.2.5.
Example 10.2.6.
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:
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.
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.
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.
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.
\(f(x)=3x- 4\)
\(\quad g(x)=x^3+2x^2 \)
\(\quad \text{Find } (f \circ g)(x) \)