Transformations and Inverses of Functions in Precalculus

Transformations of Functions

Introduction

Transformations allow us to modify the graph of a function in systematic ways, such as shifting, stretching, compressing, or reflecting it. Understanding these transformations is essential for graphing and analyzing functions in Precalculus.

Transformation of $f(x)$	Effect on the graph of $f$
$f(x) + c$	Vertical shift up by $c$ units
$f(x) - c$	Vertical shift down by $c$ units
$f(x + c)$	Shift left by $c$ units
$f(x - c)$	Shift right by $c$ units
$c f(x)$	Vertical stretch if $c > 1$; vertical compression if $0 < c < 1$
$f(cx)$	Horizontal stretch if $0 < c < 1$; horizontal compression if $c > 1$
$-f(x)$	Reflection about the $x$-axis
$f(-x)$	Reflection about the $y$-axis

Example:

• Given $f(x) = x^2$, the graph of $f(x) + 3$ is $x^2$ shifted up by 3 units.

• The graph of $f(x - 2)$ is $x^2$ shifted right by 2 units.

Graphing Trigonometric Functions

Introduction

Trigonometric functions such as sine and cosine can be transformed using amplitude, period, phase shift, and vertical shift. The general form is:

$f(x) = A \sin(B(x - a)) + C$

• Amplitude: $|A|$ (height from the center line to a peak)

• Period: $\frac{2\pi}{|B|}$ (length of one cycle)

• Horizontal shift (phase shift): $a$ units

• Vertical shift: $C$ units

Example:

• For $f(x) = 2\sin(3(x - \frac{\pi}{4})) + 1$:

  • Amplitude: 2

  • Period: $\frac{2\pi}{3}$

  • Horizontal shift: $\frac{\pi}{4}$ units right

  • Vertical shift: 1 unit up

Inverse Functions

Definition and Properties

An inverse function reverses the effect of the original function. For a function $f(x)$ with domain $D$ and range $R$, its inverse $f^{-1}(y)$ (if it exists) has domain $R$ and range $D$, and satisfies:

$f^{-1}(f(x)) = x$ for all $x$ in $D$ $f(f^{-1}(y)) = y$ for all $y$ in $R$

• A function is one-to-one if $f(x_1) \neq f(x_2)$ whenever $x_1 \neq x_2$.

• Only one-to-one functions have inverses that are also functions.

Example:

• The function $f(x) = 2x + 3$ is one-to-one and has the inverse $f^{-1}(y) = \frac{y - 3}{2}$.

Inverse Trigonometric Functions

Domain and Range of $\cos^{-1}(x)$

• Domain: $-1 \leq x \leq 1$

• Range: $0 \leq y \leq \pi$

Example:

• $\cos^{-1}(1) = 0$

• $\cos^{-1}(0) = \frac{\pi}{2}$

• $\cos^{-1}(-1) = \pi$

Practice and Application

Sample Poll Question

Let $f(x) = \sin(x)$ and consider the transformation $f(x) + c$ for $c > 0$. Which properties of the function will change?

• Amplitude: No

• y-intercept: Yes

• Period: No

Only the y-intercept and vertical position change; amplitude and period remain the same.

Summary of Learning Outcomes

• Apply transformations to functions

• Determine whether or not a function has an inverse

• Find the inverse of a function, if it exists