Precalculus Practice: Functions, Domains, Transformations, and Piecewise Functions

Functions and the Difference Quotient

Difference Quotient

The difference quotient is a fundamental concept in calculus and precalculus, used to compute the average rate of change of a function over an interval. It is defined as:

• Formula: $\frac{f(x+h) - f(x)}{h}$

• This expression approximates the slope of the secant line between two points on the graph of $f(x)$ separated by $h$ units.

Example: For $f(x) = 3x^2 + 2x$, the difference quotient is:

• Compute $f(x+h)$ and $f(x)$, then simplify $\frac{f(x+h) - f(x)}{h}$.

Solving for Parameters in Functions

Finding Unknowns

Given a function with an unknown parameter, you can solve for the parameter by substituting a known input-output pair.

• Example: If $f(x) = \frac{2x - 5}{x + B}$ and $f(1) = 3$, substitute $x = 1$ and $f(1) = 3$ to solve for $B$.

Domain of Functions

Finding the Domain

The domain of a function is the set of all real numbers for which the function is defined.

• For rational functions, exclude values that make the denominator zero.

• For functions involving square roots, ensure the radicand is non-negative.

Example: Find the domain of $\sqrt{\frac{x-1}{3x-21}}$.

• Set $3x-21 \neq 0$ and $\frac{x-1}{3x-21} \geq 0$.

• Express the domain in interval notation.

Evaluating and Analyzing Functions

Function Evaluation and Points on the Graph

To evaluate a function at a specific value, substitute the value for $x$ in the function's formula.

• Example: For $f(x) = \frac{x+3}{x+7}$, evaluate $f(1)$ and interpret the point $(1, f(1))$ on the graph.

• To find $x$ such that $f(x) = 0$, solve $\frac{x+3}{x+7} = 0$.

• The domain excludes $x = -7$ (where the denominator is zero).

Intercepts of Functions

Finding Intercepts

• x-intercept: Set $f(x) = 0$ and solve for $x$.

• y-intercept: Evaluate $f(0)$.

Transformations of Functions

Graphing with Transformations

Transformations shift, stretch, or reflect the graph of a function. Common transformations include:

• Vertical Shifts: $f(x) + c$ shifts up by $c$ units.

• Horizontal Shifts: $f(x - h)$ shifts right by $h$ units.

• Reflections: $-f(x)$ reflects over the x-axis.

• Vertical Stretch/Compression: $a f(x)$ stretches if $|a| > 1$, compresses if $0 < |a| < 1$.

Example: To graph $f(x) = -2(x-5)^2 + 4$:

• Start with $y = x^2$.

• Shift right by 5: $y = (x-5)^2$.

• Reflect and stretch: $y = -2(x-5)^2$.

• Shift up by 4: $y = -2(x-5)^2 + 4$.

Average Rate of Change

Definition and Calculation

The average rate of change of a function $g(x)$ from $x = a$ to $x = b$ is:

• Formula: $\frac{g(b) - g(a)}{b - a}$

• This measures the change in $g(x)$ per unit change in $x$ over the interval $[a, b]$.

Example: For $g(x) = -3x^2 - 2x + 3$, find the average rate of change from $x = 0$ to $x = 2$.

Piecewise Functions

Definition and Domain

A piecewise function is defined by different expressions over different intervals of the domain.

• Example:

Interval	Expression
$0 < x \leq 2$	$x$
$2 < x < 3$	$x^2$
$x \geq 3$	$7$

• To find the domain, combine the intervals where the function is defined.

Function Composition

Composing Functions

The composition of two functions $f$ and $k$ is written as $f(k(x))$ and means to substitute $k(x)$ into $f(x)$.

• Example: If $k(x) = x^2 - x$, then $f(k(x))$ is found by replacing every $x$ in $f(x)$ with $x^2 - x$.

• Similarly, for $k(x) = \sqrt{x^2 + 1}$, substitute into $f(x)$ as needed.