04/14 4. Functions and Their Graphs

Relations, Functions, and Graphical Tests

A Relation is a set of ordered pairs $x,y$, where the first coordinate is the input and the second coordinate is the output. For example, $R=\{(1,3),(2,5),(3,7)\}$ relates each input to one output.

A Function is a relation in which every input has exactly one output. Different inputs may share an output, but one input cannot be assigned two different outputs.

For example, $\{(1,4),(2,4),(3,9)\}$ is a function because each input appears once. In contrast, $\{(1,4),(1,7),(3,9)\}$ is not a function because input $1$ is paired with two outputs.

The Vertical line test provides a graph-based check: if any vertical line crosses a graph more than once, the graph is not a function of $x$, because that input would have multiple outputs.

Takeaway: A function assigns exactly one output to every permitted input, whether the relationship is shown as pairs, a table, a mapping, or a graph.

Domain and Range

The Domain is the set of possible inputs, and the Range is the set of resulting outputs. For

the domain is $\{-2,0,3,4\}$ and the range is $\{1,5,8\}$. The repeated output $5$ appears only once in the range.

From a graph, inspect the horizontal values covered to find the domain and the vertical values covered to find the range. An open endpoint is excluded, while a closed endpoint is included. For example, a graph that begins just after $x=-3$ and ends at an included $x=4$ has domain

Parentheses indicate excluded endpoints, brackets indicate included endpoints, and infinity always uses a parenthesis because it is not an included number.

Equations also impose restrictions:

• A denominator cannot equal zero. Thus, $f(x)=\frac{1}{x-2}$ has domain $x\ne 2$, or $( -\infty,2)\cup(2,\infty)$.

• The radicand of an even root must be nonnegative for a real-valued function. For $g(x)=\sqrt{x+5}$, the condition $x+5\ge 0$ gives domain $[-5,\infty)$.

• A polynomial, such as $p(x)=x^3-2x+1$, has domain all real numbers.

A real-world context can impose additional restrictions, such as allowing only nonnegative integers for the number of items sold.

Takeaway: Determine domain and range from the permitted inputs and outputs, then check both the graph and the equation for exclusions.

Function Notation and Representations

Function notation names a function and describes its output. The statement $y=f(x)$ means that the output $y$ is produced by function $f$ when the input is $x$; it does not mean $f$ multiplied by $x$.

To evaluate a function, substitute the input everywhere the variable appears. If

then

Therefore, the point $(2,9)$ lies on the graph.

For a symbolic input, substitute the entire expression. If $f(x)=x^2+4x$, then

The input is often called the independent variable because it is selected first. The output is the dependent variable because its value depends on the input.

A function can be represented in several equivalent ways:

• Verbally: the output is five more than twice the input.

• Numerically: a table of input-output values.

• Algebraically: $f(x)=2x+5$.

• Graphically: points or a curve in the coordinate plane.

• As a mapping: arrows from inputs to outputs.

For $f(x)=2x+5$, inputs $-2,-1,0,1$ produce outputs $1,3,5,7$, respectively. A table is especially useful for discrete data, while an equation or graph can describe a continuous relationship. Each representation must still assign exactly one output to each permitted input.

Takeaway: Substitute carefully in function notation, and use multiple representations to reveal the same input-output rule from different perspectives.

Transformations of Function Graphs

A Transformation changes the graph of a known parent function in a predictable way. For a parent function $y=f(x)$, the main rules are:

• $g(x)=f(x)+k$ shifts the graph up $k$ units when $k>0$; $g(x)=f(x)-k$ shifts it down $k$ units.

• $g(x)=f(x-h)$ shifts the graph right $h$ units, while $g(x)=f(x+h)$ shifts it left $h$ units. The horizontal sign works opposite to the direction of movement.

• $g(x)=-f(x)$ reflects the graph across the $x$-axis, while $g(x)=f(-x)$ reflects it across the $y$-axis.

• $g(x)=af(x)$ changes vertical distances by a factor of $|a|$. Values $|a|>1$ produce a vertical stretch, while $0<|a|<1$ produce a vertical compression.

• $g(x)=f(bx)$ changes horizontal distances by a factor of $1/|b|$. Values $|b|>1$ produce a horizontal compression, while $0<|b|<1$ produce a horizontal stretch.

A combined form is

Here, $h$ controls horizontal position, $k$ controls vertical position, $a$ controls vertical scaling and possible reflection, and $b$ controls horizontal scaling and possible reflection.

For example, with $f(x)=|x|$ and $g(x)=-2|x-3|+1$:

1. $x-3$ shifts the graph right $3$ units.

2. The factor $2$ stretches it vertically by a factor of $2$.

3. The negative sign reflects it across the $x$-axis.

4. The addition of $1$ shifts it up $1$ unit.

The vertex moves from $(0,0)$ to $(3,1)$.

Takeaway: Read transformations from the outside and inside of the function, remembering that horizontal shifts have an opposite sign inside the input.

Inverse Relations and Functions

An Inverse relation reverses ordered pairs. If

then

The graph is reflected across $y=x$, since every point $x,y$\) becomes $y,x$\).

An Inverse function exists only when the original function is one-to-one. Use the Horizontal line test: if some horizontal line intersects the graph more than once, two different inputs produce the same output, so the inverse relation is not a function.

For example, $f(x)=x^2$ is not one-to-one on all real numbers because

If its domain is restricted to $[0,\infty)$, it becomes one-to-one and its inverse is $f^{-1}(x)=\sqrt{x}$.

To find an inverse algebraically:

1. Write $y=f(x)$.

2. Interchange $x$ and $y$.

3. Solve for $y$.

4. Rename the result $f^{-1}(x)$.

For $f(x)=3x-7$, interchange the variables in $y=3x-7$ to obtain $x=3y-7$. Solving gives

Inversion exchanges domain and range:

The identities

provide checks on the appropriate domains.

Takeaway: Reverse the pairs, reflect across $y=x$, and verify one-to-one behavior before treating an inverse relation as a function.

• One-to-one function