Algebra

Functions

Explore inputs, inverses and composition.

Predict, test and explain

Step 1 of 3

Predict

For f(x) = 2x + 3, what is f(2)?

f(x)=2x+3

Scroll the diagram sideways to see every label.

f(2)=7

A function assigns one output to each input. Change the multiplier, constant or input.

Function journey
Stage	Value
Input	2
Output f(x)	7

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

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Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

Find the composite output shown below.

$f(x)=2x+1,\quad g(x)=x-3;\qquad f(g(5))$

Hint

Apply the inner function first, then feed its result into the outer function.

Reveal answer

The inner function gives two; doubling it and adding one gives five.

$f(g(5))=f(2)=5$

Hint 1

A function gives one output for each permitted input. f(x) names the output; it does not mean f multiplied by x.

Hint 2

An inverse function reverses the original function. For an affine function ax + b with a ≠ 0, undo addition and multiplication in reverse order.

Hint 3

In f(g(x)), apply g first and feed its output into f. Reversing the order can change the result.

Worked example

This example uses fixed values, separate from the diagram controls.

Let f(x) = 2x + 3 and g(x) = x − 1. Find f⁻¹(x), f(g(x)) and g(f(x)).

f doubles its input and adds 3. To reverse it, subtract 3 and divide by 2: f⁻¹(x) = (x − 3)/2.
For f(g(x)), put x − 1 into f: 2(x − 1) + 3 = 2x + 1.
For g(f(x)), put 2x + 3 into g: (2x + 3) − 1 = 2x + 2.
The two composite functions differ because the operations were applied in a different order.

Answer: f⁻¹(x) = (x − 3)/2; f(g(x)) = 2x + 1; g(f(x)) = 2x + 2.

Connect this idea

Watch out: f⁻¹(x) is an inverse function, not 1/f(x). A constant affine function has no inverse on all real inputs.