Functions, graphs, and set relationships

This section covers how to evaluate functions, graph lines, and read key features of a line, such as slope and intercepts. It also covers common linear equation forms - slope-intercept, point-slope, and standard - and uses the intercept method to graph equations in standard form. It closes with set relationships - unions, intersections, and Venn diagrams - and word problems.

Evaluating functions

A function is a rule that pairs each input with exactly one output. Evaluating a function means finding the output for a specific input. The notation $f(a)$ means “the value of the function when $x=a$.”

The main skill is substitution:

• Replace every $x$ in the formula with the given input value.
• Use parentheses around the substituted value.
• Simplify using the order of operations.

Parentheses matter most when the input is negative or when the expression includes exponents or fractions.

Watch out with negative inputs: always wrap a negative input in parentheses before applying exponents. For example, $(-2{)}^2=4$, not $-4$. Dropping the parentheses is one of the most common errors on function evaluation problems.

Example: evaluate a quadratic function with a negative input

Let $f(x)=2x^2-3x+5$. Find $f(-2)$.

• Substitute, keeping parentheses around $-2$: $f(-2)=2(-2{)}^2-3(-2)+5$.
• Compute each term: $2(4)+6+5=8+6+5=19$.

Answer: $19$

Graphing linear functions

Graphing a linear function shows all solutions to the equation. Each point on the graph is an ordered pair $(x,y)$ that satisfies the equation, and all of those points lie on a straight line.

Because a linear function has a constant rate of change, its graph extends infinitely in both directions unless the problem states a restriction.

The method you use depends on the form of the equation:

• Some forms make slope and intercepts easy to read.
• Other forms are more convenient when you’re given a point on the line.

Being able to move between equations, tables, and graphs makes these questions much faster.

Slope-intercept form

Use slope-intercept form when you can read the slope $m$ and $y$-intercept $b$ directly from the equation. It’s the most convenient form for quickly sketching a line.

For $f(x)=mx+b$:

1. Plot the $y$-intercept $(0,b)$.
2. Use slope $m=\frac{\Delta y}{\Delta x}$ to find a second point.
3. Draw the line through both points.

Example: graph $y=2x+1$

• $y$-intercept: $(0,1)$.
• Slope $2=\frac{2}{1}$, so from $(0,1)$ go right $1$ and up $2$ to $(1,3)$.

Answer: See graph

Horizontal and vertical lines

Two special cases come up often on the Praxis and are easy to confuse.

Horizontal lines have the equation $y=b$, where $b$ is a constant. Every point on the line has the same $y$-value, so the slope is $0$ (zero rise, any run). For example, $y=3$ is a horizontal line crossing the $y$-axis at $(0,3)$.

Vertical lines have the equation $x=a$, where $a$ is a constant. Every point on the line has the same $x$-value. Moving from one point to another involves zero run, so slope $=\frac{\text{rise}}{0}$, which is undefined.

Quick reference:

• $y=b$ → horizontal line, slope $=0$
• $x=a$ → vertical line, slope is undefined

Point-slope form

Use point-slope form when you know a point on the line and the slope, but the point isn’t the $y$-intercept. It’s especially handy when you derive a line from two given points (see the next section).

The form

$$y-y_1=m(x-x_1)$$

builds the line around the anchor point $(x_1,y_1)$.

When you know $(x_1,y_1)$ and slope $m$:

1. Write $y-y_1=m(x-x_1)$.
2. Use the slope to plot additional points, or rearrange into slope-intercept form if needed.

Finding a line from two points

When a problem gives you two points but no slope, you need to calculate the slope first, then use point-slope form to write the equation.

Step 1: Compute the slope using the two points $(x_1,y_1)$ and $(x_2,y_2)$:

$$m=\frac{y_2-y_1}{x_2-x_1}$$

Keep the order consistent - use the same point as “point 2” in both the numerator and denominator, or you’ll flip the sign of the slope.

Step 2: Substitute $m$ and either point into point-slope form:

$$y-y_1=m(x-x_1)$$

Step 3: Simplify into slope-intercept form if needed, then graph using the slope and one of the points.

Example: find the equation of a line through two points

A line passes through $(2,-1)$ and $(3,2)$.

• Compute the slope: $m=\frac{2-(-1)}{3-2}=\frac{3}{1}=3$.
• Substitute into point-slope form using $(2,-1)$: $y-(-1)=3(x-2)$.
• Simplify: $y+1=3x-6$, so $y=3x-7$.
• To graph, plot $(2,-1)$ and use the slope $3=\frac{3}{1}$ to move right $1$ and up $3$ to $(3,2)$.

Answer: $y=3x-7$

Standard form and intercept method

Use standard form when the equation is already written as $Ax+By=C$ and you want to graph it quickly without rearranging - the intercept method gets you two points with minimal algebra.

The idea is to find where the line crosses each axis:

• The $x$-intercept occurs where the graph crosses the $x$-axis, which happens when $y=0$.
• The $y$-intercept occurs where the graph crosses the $y$-axis, which happens when $x=0$.

Once you have both intercepts, plot them and draw the line through the two points.

Example: find the intercepts of $2x+3y=6$

• Set $y=0$ to find the $x$-intercept: $2x=6\Rightarrow x=3$, so $(3,0)$.
• Set $x=0$ to find the $y$-intercept: $3y=6\Rightarrow y=2$, so $(0,2)$.

Answer: $(3,0)$ and $(0,2)$

Set relationships and word problems

A set is a collection of distinct objects. Two operations describe how sets overlap:

Definitions

Union ($A\cup B$)

The set containing all elements that are in $A$, or in $B$, or in both.

Intersection ($A\cap B$)

The set containing all elements that are in both $A$ and $B$.

Parentheses control the order of set operations the same way they control arithmetic. In an expression like $(A\cup B)\cap C$, find the union inside the parentheses first, then intersect the result with $C$.

Example: combined set operations

Find $(A\cup B)\cap C$ where $A=\{1,2,3\}$, $B=\{3,4,5\}$, and $C=\{2,3,5,7\}$.

• First, $A\cup B=\{1,2,3,4,5\}$.
• Then, $(A\cup B)\cap C=\{2,3,5\}$.

Answer: $\{2,3,5\}$

Venn diagrams

A Venn diagram uses overlapping circles to represent sets. The overlap shows the intersection (elements in both sets), and the non-overlapping parts show elements that belong to only one set.

To count the elements in each region of a two-set Venn diagram, use the inclusion-exclusion formula:

$$n(A\cup B)=n(A)+n(B)-n(A\cap B)$$

Here $n(A\cup B)$ is the number of elements in either set, and $n(A\cap B)$ is the number in both. Subtracting the overlap keeps you from counting the shared elements twice. Plug in what you know, solve for what you don’t, then fill in each region.

Example: math and science

In a group of $100$ students, $60$ like math, $40$ like science, and $20$ like both. How many like only math, only science, and neither?

Let $M$ = students who like math ($n(M)=60$) and $S$ = students who like science ($n(S)=40$), with $n(M\cap S)=20$.

• Inclusion-exclusion: $n(M\cup S)=60+40-20=80$
• Both: $20$
• Only math: $60-20=40$
• Only science: $40-20=20$
• Neither: $100-80=20$

Answer: only math $40$, only science $20$, neither $20$

Example: fair attractions

At a fair, $100$ people tried rides, $75$ tried games, and $40$ tried both. How many tried exactly one?

• $n(A\cup B)=100+75-40=135$
• Rides only: $100-40=60$; Games only: $75-40=35$; Exactly one: $60+35=95$

Answer: $95$

Linear relationships in word problems

Word problems that describe a linear relationship ask you to translate the situation into a function, then substitute the given value.

Example: linear relationship word problem

A rental company charges a $25 base fee plus $0.40 per mile. Write a function for the total cost $C$ in terms of miles driven $m$, and find the cost of driving $60$ miles.

• Translate the situation into a function: $C(m)=0.40m+25$.
• Substitute $m=60$: $C(60)=0.40(60)+25=49$.

Answer: $49