Functions and Graphs

What you will learn

• Use function notation f(x).
• Find domain and range, composite and inverse functions.
• Recognise the graph of a function.

Before you start: you should know substitution and straight-line graphs. See Linear Equations.

The idea

A function is a machine: put a number in, apply a rule, get a number out. The crucial condition is that each input produces exactly one output. The rule “double it and add one” is a function; so is “square it”.

We name functions with letters and write

$f(x) = 2x + 1$

read as ”$f$ of $x$ equals $2x + 1$”. Then $f(3)$ means “put $3$ into the machine”: $f(3) = 2(3) + 1 = 7$.

Domain and range

• The domain is the set of inputs allowed. For $f(x) = \sqrt{x}$, the domain is $x \geq 0$, because you cannot take the real square root of a negative number.
• The range is the set of outputs the function actually produces. For $g(x) = x^2$, the range is $g(x) \geq 0$.

Graphs

Plot every input $x$ against its output $f(x)$ and you get the graph $y = f(x)$. The graph is a complete visual record of the function. A quick test: a curve is the graph of a function only if every vertical line crosses it at most once.

Transforming graphs

Small changes to the rule move the graph in predictable ways:

• $f(x) + 3$ shifts the graph up $3$.
• $f(x - 3)$ shifts it right $3$ (note the opposite sign).
• $2f(x)$ stretches it vertically by factor $2$.
• $-f(x)$ reflects it in the $x$-axis.

Worked examples

Example 1: Evaluating and solving

For $f(x) = x^2 - 4x$, find $f(-2)$ and solve $f(x) = 5$.

1. Substitute: $f(-2) = (-2)^2 - 4(-2) = 4 + 8 = 12$.
2. To solve $f(x) = 5$, write $x^2 - 4x = 5$, so $x^2 - 4x - 5 = 0$.
3. Factorise: $(x - 5)(x + 1) = 0$.

Answer: $f(-2) = 12$; $f(x) = 5$ when $x = 5$ or $x = -1$.

Example 2: Composite functions

Let $f(x) = 3x + 1$ and $g(x) = x^2$. Find $f(g(2))$ and $g(f(2))$.

1. Work from the inside out. $g(2) = 4$, then $f(4) = 13$.
2. For the other order: $f(2) = 7$, then $g(7) = 49$.

Answer: $f(g(2)) = 13$ and $g(f(2)) = 49$. Order matters.

Example 3: Finding an inverse

Find the inverse of $f(x) = \dfrac{x + 5}{2}$.

1. Write $y = \dfrac{x + 5}{2}$.
2. Rearrange to make $x$ the subject: $2y = x + 5$, so $x = 2y - 5$.
3. Swap the letters back to name the inverse: $f^{-1}(x) = 2x - 5$.
4. Check: $f(3) = 4$ and $f^{-1}(4) = 3$. ✓

Answer: $f^{-1}(x) = 2x - 5$

The graph of an inverse function is the reflection of the original graph in the line $y = x$.

Why it matters

Functions are how mathematics models the world. Temperature as a function of time, profit as a function of price, the height of a ball as a function of seconds since it was thrown. Every spreadsheet column calculated from another is a function, and every line of code that takes an input and returns a value is one too.

Functions are the objects that calculus studies. The derivative of a function describes its rate of change, and composite functions are exactly what the chain rule differentiates.

Practice

Your turn. Type or choose an answer, then press Check. Getting one wrong is fine, that is what practice is for. You can try as many times as you like.

Question 1 of 8

$f(x) = 2x + 3$. Find $f(4)$.

Your answer

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Question 2 of 8

$f(x) = x^2 - 1$. Find $f(-3)$.

Your answer

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Question 3 of 8

$f(x) = x + 2$ and $g(x) = 3x$. Find $f(g(2))$.

Your answer

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Question 4 of 8

With the same $f$ and $g$, find $g(f(2))$.

Your answer

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Question 5 of 8

$f(x) = 2x + 3$. Solve $f(x) = 11$.

Your answer

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Question 6 of 8

$f(x) = x + 5$. What is $f^{-1}(10)$?

Your answer

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Question 7 of 8

Which value must be excluded from the domain of $f(x) = \dfrac{1}{x}$?

Choose one answer

01-110

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Question 8 of 8

What is the range of $f(x) = x^2$?

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$f(x) \ge 0$all numbers$f(x) > 0$$f(x) \le 0$

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