Functions

Functions

The language of functions

A function takes each input to exactly one output.

$\mathrm{f}(x) = x^2$ for all real $x$ is many-one, because $\mathrm{f}(2) = \mathrm{f}(-2)$. With the domain $x \geq 0$ it becomes one-one.

Finding the range

Sketch the graph over the given domain, or complete the square for a quadratic, and read off the $y$-values.

$\mathrm{f}(x) = (x - 2)^2 + 3$, and the bracket is never negative, so $\mathrm{f}(x) \geq 3$.

On this domain the function is increasing, since the turning point at $x = 2$ is outside it. $\mathrm{f}(3) = 4$ and $\mathrm{f}(5) = 12$, so $4 \leq \mathrm{f}(x) \leq 12$.

Composite functions

$\mathrm{fg}(x)$ means $\mathrm{f}(\mathrm{g}(x))$: apply $\mathrm{g}$ first, then $\mathrm{f}$.

$\mathrm{fg}(x) = 2x^2 + 3 \qquad \mathrm{gf}(x) = (2x + 3)^2$

They are different. Order matters.

Inverse functions

Only a one-one function has an inverse. The inverse undoes the function, so $\mathrm{f}^{-1}\mathrm{f}(x) = x$.

To find it: write $y = \mathrm{f}(x)$, make $x$ the subject, then swap the letters.

$y = (x - 1)^2 + 2 \quad\Rightarrow\quad x - 1 = \pm\sqrt{y - 2}$

The domain $x \geq 1$ means $x - 1 \geq 0$, so take the positive root. Then

$\mathrm{f}^{-1}(x) = 1 + \sqrt{x - 2}, \quad x \geq 2$

$y(x - 3) = 2x + 1 \quad\Rightarrow\quad xy - 2x = 1 + 3y \quad\Rightarrow\quad x = \frac{3y + 1}{y - 2}$

So $\mathrm{f}^{-1}(x) = \dfrac{3x + 1}{x - 2}$, $x \neq 2$.

Graphs of a function and its inverse

The graph of $y = \mathrm{f}^{-1}(x)$ is the reflection of $y = \mathrm{f}(x)$ in the line $y = x$. Any point where the two meet on that line satisfies $\mathrm{f}(x) = x$.

Making a function one-one

A quadratic is one-one only on one side of its turning point.

The turning point is at $x = 2$, since $\mathrm{f}(x) = (x - 2)^2 - 3$. So the least value is $k = 2$.

Transformations of graphs

Combining transformations

Apply the changes inside the bracket first, then those outside. Outside the bracket, a multiplication comes before an addition, just as in arithmetic.

1. Translation by $\begin{pmatrix} 2 \\ 0 \end{pmatrix}$, giving $y = (x - 2)^2$.
2. Stretch parallel to the $y$-axis, scale factor 3, giving $y = 3(x - 2)^2$.
3. Translation by $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$.

Reflecting gives $y = -\mathrm{f}(x)$. Translating gives $y = 4 - \mathrm{f}(x)$. Doing them in the other order gives $y = -\mathrm{f}(x) - 4$, which is different.