ACT Math 4: Intermediate Algebra — Functions, Matrices, and Logarithms

Relation

A relation is a set of ordered pairs. The domain is the set of first components, and the range is the set of second components.

What are the domain and range of the relation $\{(1,2),(2,4),(3,6),(4,8),(5,10)\}$?

The domain is $\{1,2,3,4,5\}$, and the range is $\{2,4,6,8,10\}$.

Function

A function is a relation in which every input in the domain is assigned exactly one output in the range. Equivalently, no input value is paired with two different outputs.

How can you determine from a set of ordered pairs whether the relation is a function?

List or inspect the input values. If any input is paired with more than one output, the relation is not a function; otherwise, it is a function.

Can two different inputs in a function have the same output?

Yes. A function restricts each input to one output, but multiple inputs may produce the same output.

Why is $\{(\text{odd},1),(\text{odd},3),(\text{odd},5)\}$ not a function?

The single input "odd" is paired with three different outputs: $1$, $3$, and $5$. This violates the requirement that each input have exactly one output.

Independent and dependent variables

The independent variable is the input and is often denoted by $x$. The dependent variable is the output and is often denoted by $y$, because its value depends on the input.

If every menu item has exactly one price, is price a function of item? Is item necessarily a function of price?

Price is a function of item. Item is not necessarily a function of price because different items may have the same price.

What does the notation $y=f(x)$ mean?

It defines a function named $f$ whose input is $x$ and whose output is $y$, also written as $f(x)$.

What is the meaning of the parentheses in $f(a)$?

The parentheses indicate that $a$ is the input to the function $f$; they do not indicate multiplication.

How should $f(a+b)$ be interpreted?

First calculate the input expression $a+b$, then use that result as the input to $f$. It does not generally mean $f(a)+f(b)$.

What does $f(2005)=300$ mean if $N=f(y)$ gives the number of police officers in year $y$?

In the year $2005$, the town had $300$ police officers.

Can function inputs be nonnumeric?

Yes. Inputs can be names, labels, or other objects, although algebraic functions typically use numerical inputs and outputs.

How can a function be represented in a table?

A table lists corresponding input and output values. It represents a function if no input appears with more than one output.

How do you evaluate a function given by a table?

Find the specified input in the input row or column, then read the output paired with it.

How do you solve $g(n)=c$ using a table?

Find every occurrence of the output value $c$ in the table and list all input values paired with it. There may be more than one solution.

A table gives $g(1)=8$, $g(2)=6$, $g(3)=7$, $g(4)=6$, and $g(5)=8$. What are the solutions to $g(n)=6$?

The solutions are $n=2$ and $n=4$, because both inputs produce the output $6$.

How do you evaluate an algebraic function such as $f(x)=x^2+3x-4$ at a given input?

Substitute the input value for every occurrence of $x$, then simplify the resulting expression.

Evaluate $f(2)$ for $f(x)=x^2+3x-4$.

$f(2)=2^2+3(2)-4=6$.

What is $f(a+h)$ for $f(x)=x^2+3x-4$?

$f(a+h)=(a+h)^2+3(a+h)-4=a^2+2ah+h^2+3a+3h-4$.

Simplify $\frac{f(a+h)-f(a)}{h}$ for $f(x)=x^2+3x-4$, assuming $h\ne0$.

Using substitution and factoring, $\frac{f(a+h)-f(a)}{h}=2a+h+3$.

How is evaluating $f(c)$ different from solving $f(x)=c$?

Evaluating $f(c)$ finds one output for a known input. Solving $f(x)=c$ finds every input that produces a specified output, so multiple solutions are possible.

For $h(p)=p^2+2p$, what inputs satisfy $h(p)=3$?

Solve $p^2+2p=3$, giving $(p+3)(p-1)=0$. Thus, $p=-3$ or $p=1$.

How can an equation be rewritten as an explicit formula for one variable as a function of another?

Algebraically isolate the output variable on one side, leaving an expression involving only the input variable on the other side.

Rewrite $2n+6p=12$ as $p=f(n)$.

Solving for $p$ gives $p=2-\frac{1}{3}n$, so $p=f(n)=2-\frac{1}{3}n$.

Why does $x^2+y^2=1$ not define $y$ as a single-valued function of $x$?

Solving for $y$ gives $y=\pm\sqrt{1-x^2}$. For most allowable $x$-values, there are two possible outputs, so it is not a function $y=f(x)$ over the full circle.

Explicit rule versus implicit rule for a function

An explicit rule gives the output directly as a formula involving the input, such as $y=2x+1$. An implicit relationship may determine a unique output without providing a simple explicit formula.

How do you read a function value from a graph?

Locate the input on the horizontal axis, move vertically to the graph, and read the corresponding vertical coordinate. That coordinate is the output $f(x)$.

How do you solve $f(x)=k$ from a graph?

Find where the horizontal line $y=k$ intersects the graph. The $x$-coordinates of all intersection points are the solutions.

If a graph passes through $(2,1)$, what is $f(2)$?

$f(2)=1$, because the point's input coordinate is $2$ and its output coordinate is $1$.

One-to-one function

A one-to-one function is a function in which each output corresponds to exactly one input. Thus, distinct inputs cannot produce the same output.

Is the area of a circle a one-to-one function of its positive radius?

Yes. Since $A=\pi r^2$ and $r>0$, each radius gives one area and each positive area corresponds to the unique radius $r=\sqrt{A/\pi}$.

Vertical line test

A graph represents a function of $x$ if every vertical line intersects it at most once. If any vertical line intersects the graph more than once, one input has multiple outputs and the graph is not a function.

Horizontal line test

A function is one-to-one if every horizontal line intersects its graph at most once. More than one intersection means one output corresponds to multiple inputs.

Why can solving a function equation produce several input values even though the function gives only one output per input?

The function rule restricts outputs for each individual input, but different inputs may share the same output. Therefore, an equation such as $f(x)=k$ can have multiple solutions.

What coordinates make up the graph of $y=f(x)$?

The graph is the set of all points $(x,y)$ satisfying $y=f(x)$, where $x$ is an input and $y$ is its corresponding output.