Functions Reference

Functions

A function is a rule that assigns each input exactly one output. Function notation lets us evaluate, combine, transform, and reverse those rules.

Solved Examples

Jump directly to a worked example by function type.

Fact Table

IdeaMeaningWatch For
FunctionEach input has exactly one output.A vertical line can hit the graph only once.
DomainAllowed inputs.No zero denominators or even roots of negatives.
RangePossible outputs.Read the $y$-values from the graph or rule.
CompositionOne function inside another.Work from the inside outward.
InverseUndoes a function.Swap $x$ and $y$, then solve for $y$.
TransformationMoves or changes a graph.Outside changes affect $y$; inside changes affect $x$.

Content Formulas

Function Notation
$$f(a)=\text{output when }x=a$$
Composition
$$(f\circ g)(x)=f(g(x))$$
Inverse Test
$$f(f^{-1}(x))=x,\quad f^{-1}(f(x))=x$$
Transformation Form
$$y=a f(x-h)+k$$
A relation can fail to be a function, but a function can still have repeated outputs. The rule is one output per input, not one input per output.

Classic Examples

Evaluate a Function

If $f(x)=2x^2-3x+1$, find $f(-2)$.

Solution Steps
  1. Replace every $x$ with the input $-2$.
  2. Evaluate the powers and products.
  3. State the function value: $f(-2)=15$.
$$f(-2)=2(-2)^2-3(-2)+1$$$$f(-2)=8+6+1$$$$f(-2)=15$$

Find the Domain

Find the domain of $f(x)=\dfrac{3}{x-5}$.

Solution Steps
  1. Require the denominator to be nonzero.
  2. Find the excluded input.
  3. State the domain in interval notation.
$$x-5\ne 0$$$$x\ne 5$$$$\text{Domain: }(-\infty,5)\cup(5,\infty)$$

Compose Functions

If $f(x)=x^2+1$ and $g(x)=3x-2$, find $(f\circ g)(x)$.

Solution Steps
  1. Write composition as $f(g(x))$.
  2. Substitute the entire inside function into $f$.
  3. Expand the squared binomial.
  4. Combine like terms and state the composition.
$$(f\circ g)(x)=f(g(x))$$$$=(3x-2)^2+1$$$$=9x^2-12x+4+1$$$$=9x^2-12x+5$$

Find an Inverse

Find the inverse of $f(x)=2x-7$.

Solution Steps
  1. Write the function as $y=2x-7$.
  2. Swap $x$ and $y$.
  3. Isolate $y$.
  4. State the inverse function.
$$y=2x-7$$$$x=2y-7$$$$x+7=2y$$$$f^{-1}(x)=\frac{x+7}{2}$$

Read a Transformation

Describe $g(x)=-2f(x-3)+4$.

Solution Steps
  1. Read $x-3$ as a shift right by $3$.
  2. Read $-2$ as a reflection over the x-axis and a vertical stretch by $2$.
  3. Read $+4$ as a shift up by $4$.
$x-3$ means right $3$.$-2$ means reflect over the x-axis and stretch by $2$.$+4$ means up $4$.