Practice Function Notation in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Function notation f(x)f(x) is a shorthand that names a function (ff) and specifies its input (xx). Writing f(3)=10f(3) = 10 means that when the input is 3, the function produces the output 10. This notation is not multiplication.

The notation f(x)f(x) is not "ff times xx" โ€” it means "the output of function ff when the input is xx." The parentheses contain the input, not a multiplication.

Showing a random 20 of 50 problems.

Example 1

medium
If f(x)=x2โˆ’4xf(x) = x^2 - 4x, find all xx with f(x)=0f(x) = 0.

Example 2

medium
If f(x)=x2f(x) = x^2, compute f(a)โˆ’f(b)f(a) - f(b) in factored form.

Example 3

challenge
A function satisfies f(xy)=f(x)+f(y)f(xy) = f(x) + f(y) for all positive x,yx, y and f(2)=1f(2) = 1. Find f(8)f(8).

Example 4

challenge
If f(x)=xx+1f(x) = \frac{x}{x+1}, simplify f(f(x))f(f(x)).

Example 5

medium
If f(x)=2xโˆ’1f(x) = 2x - 1 and g(x)=x2+3g(x) = x^2 + 3, find (fโˆ˜g)(x)(f \circ g)(x) and (gโˆ˜f)(x)(g \circ f)(x).

Example 6

easy
If f(x)=2xf(x) = 2x, find f(โˆ’3)f(-3).

Example 7

easy
If f(x)=x+1f(x) = x + 1, find f(a)f(a).

Example 8

medium
If f(x)=2x+1f(x) = 2x+1 and g(x)=x2g(x) = x^2, find f(g(2))f(g(2)).

Example 9

medium
If f(x)=x2โˆ’3xf(x) = x^2 - 3x, find f(xโˆ’2)f(x-2) and simplify.

Example 10

hard
If f(x)=2x+5f(x) = 2x + 5, find fโˆ’1(x)f^{-1}(x) (the inverse).

Example 11

medium
If f(x)=x+3xโˆ’5f(x) = \frac{x+3}{x-5}, list all xx where ff is undefined.

Example 12

medium
If f(x)=4xโˆ’1f(x) = 4x - 1, find f(2x)f(2x).

Example 13

medium
If f(x)=โˆฃxโˆฃf(x) = |x|, find f(โˆ’5)+f(5)f(-5) + f(5).

Example 14

easy
If f(x)=3x2โˆ’2x+1f(x) = 3x^2 - 2x + 1, find f(โˆ’2)f(-2).

Example 15

easy
If f(x)=5f(x) = 5, find f(100)f(100).

Example 16

hard
If f(x)=1xf(x) = \frac{1}{x} for xโ‰ 0x \ne 0, simplify f(x+h)โˆ’f(x)h\frac{f(x+h) - f(x)}{h}.

Example 17

medium
If f(x)=5xโˆ’3f(x) = 5x - 3 and f(c)=22f(c) = 22, find cc.

Example 18

easy
If f(x)=โˆ’x+4f(x) = -x + 4, find f(โˆ’2)f(-2).

Example 19

easy
If f(x)=x2f(x) = x^2, find f(0)f(0).

Example 20

medium
If f(x)=3x+2f(x) = 3x + 2 and g(x)=xโˆ’1g(x) = x - 1, find g(f(1))g(f(1)).
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