Practice Input-Output View in Math

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

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Quick Recap

The input-output view of a function treats it as a black box: put in a value (input), get out a uniquely determined value (output), without worrying about the internal mechanism.

Like a vending machine: put in selection (input), get out snack (output).

Showing a random 20 of 50 problems.

Example 1

easy
Think of f(x)=3x−7f(x) = 3x - 7 as a machine. Describe the sequence of operations applied to input xx, then evaluate f(5)f(5) and find the input that gives output 1414.

Example 2

challenge
Machines: f(x)=2xf(x)=2x and g(x)=x+3g(x)=x+3. Find an input xx with f(g(x))=g(f(x))f(g(x))=g(f(x)).

Example 3

medium
Two machines: ff doubles, gg adds 3. Find g(f(2))g(f(2)).

Example 4

easy
Must the inputs of a function be numbers?

Example 5

challenge
Given f(x)=2x+3f(x) = 2x + 3, find a function gg so that f(g(x))=xf(g(x)) = x.

Example 6

medium
For f(x)=4x−9f(x) = 4x - 9, find the input that gives output −1-1.

Example 7

easy
Is ff the same as f(3)f(3)?

Example 8

easy
Does f(x)f(x) mean ff multiplied by xx?

Example 9

medium
If f(x)=1x−2f(x)=\dfrac{1}{x-2}, which input is NOT allowed, and why (input-output view)?

Example 10

easy
For f(x)=x+4f(x) = x + 4, fill in the table: f(0),f(1),f(−2)f(0), f(1), f(-2).

Example 11

medium
If f(3)=7f(3)=7 and f(5)=7f(5)=7, is ff a function? Is it one-to-one?

Example 12

easy
A function maps each student to their birth year. Is birth year the input or the output?

Example 13

easy
In f(7)=12f(7)=12, which number is the input and which is the output?

Example 14

easy
If gg squares its input, what is g(4)g(4)?

Example 15

medium
For f(x)=3−2xf(x) = 3 - 2x, find f(a+1)f(a + 1).

Example 16

medium
If f(x)=2x+1f(x) = 2x + 1 and g(x)=x−3g(x) = x - 3, find f(g(5))f(g(5)).

Example 17

hard
If f(x)=3x+1f(x) = 3x + 1, find the inverse machine that recovers xx from f(x)f(x).

Example 18

medium
If f(x)=ax+bf(x) = ax + b and f(2)=5f(2) = 5, f(4)=11f(4) = 11, find aa and bb.

Example 19

medium
For f(x)=3x−2f(x)=3x-2, find the input that produces output 1010.

Example 20

easy
Can two different inputs share the same output in a function?
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