How do you evaluate a function like f(x) on the GRE?

Substitute the given input value for every occurrence of the variable, then simplify using order of operations. For a composition f(g(x)), evaluate the inner function g first, then apply f to that result — the order cannot be swapped.

How do you find the slope of a line between two points?

Use m = (y₂ - y₁) / (x₂ - x₁), the rise over the run. Keep the point order consistent in both the numerator and denominator. A horizontal line has slope 0, and a vertical line has an undefined slope.

What is the slope of a line perpendicular to another line?

A perpendicular slope is the negative reciprocal of the original slope: flip the fraction and change its sign so the two slopes multiply to -1. For example, the perpendicular of 2/5 is -5/2, and the perpendicular of -3 is 1/3.

How do you find the distance between two points?

Use the distance formula d = √((x₂ - x₁)² + (y₂ - y₁)²), which is the Pythagorean theorem applied to the horizontal and vertical gaps between the points. Do not forget the square root at the end.

CramClub • Functions and Coordinate Geometry — GRE® General Test Guide & Practice

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

$y = mx + b$ ; $\;y - y_1 = m(x - x_1)$

$m_1 \cdot m_2 = -1$

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

$M = \left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)$

Functions and the Coordinate Plane

Two closely linked ideas power a large share of GRE algebra: functions (rules that turn an input into a single output) and coordinate geometry (the algebra of points, lines, and distances in the $xy$ -plane). The GRE tests them together — a line is a linear function, and a graph is a function you read with your eyes. This guide covers evaluation, domain and range, slope, the equation of a line, distance and midpoint, and the perpendicular/parallel rules.

Function Notation and Evaluation

The notation $f(x)$ names a rule and its input. To evaluate, substitute the input everywhere $x$ appears. If $f(x) = 2x^2 - 3x + 1$ , then

$f(4) = 2(4)^2 - 3(4) + 1 = 32 - 12 + 1 = 21.$

Composition means feeding one output into another function: $f(g(x))$ evaluates $g$ first, then $f$ . If $g(x) = x + 5$ , then $f(g(2)) = f(7)$ , and you evaluate $f$ at $7$ . Work from the inside out, and never assume $f(g(x)) = g(f(x))$ — order matters.

Domain and Range Basics

The domain is the set of allowed inputs; the range is the set of possible outputs. For most GRE functions the domain is "all real numbers," but two situations restrict it:

Division by zero is forbidden. For $f(x) = \frac{1}{x - 3}$ , the domain excludes $x = 3$ .
Even roots of negatives are not real. For $f(x) = \sqrt{x - 2}$ , you need $x - 2 \ge 0$ , so the domain is $x \ge 2$ .

Slope: The Rate of Change

The slope $m$ of a line measures how steeply it rises. Given two points $(x_1, y_1)$ and $(x_2, y_2)$ :

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

A positive slope rises left-to-right; a negative slope falls. A horizontal line has slope $0$ ; a vertical line has an undefined slope (its run is zero). The line through $(1, 2)$ and $(4, 11)$ has slope $\frac{11 - 2}{4 - 1} = \frac{9}{3} = 3$ .

The Equation of a Line

The two forms you need are slope-intercept and point-slope:

$y = mx + b \qquad\qquad y - y_1 = m(x - x_1)$

Here $b$ is the $y$ -intercept (where $x = 0$ ). To find the $x$ -intercept, set $y = 0$ and solve. For $y = 3x - 12$ , the $y$ -intercept is $-12$ and the $x$ -intercept is where $0 = 3x - 12$ , i.e. $x = 4$ .

Parallel and Perpendicular Slopes

Relationship	Slopes
Parallel lines	Equal slopes: $m_1 = m_2$
Perpendicular lines	Negative reciprocals: $m_1 \cdot m_2 = -1$

A line perpendicular to $y = \tfrac{2}{5}x + 1$ has slope $-\tfrac{5}{2}$ , because $\tfrac{2}{5}$ and $-\tfrac{5}{2}$ multiply to $-1$ .

Distance and Midpoint

Two more formulas complete the toolkit. The distance between $(x_1, y_1)$ and $(x_2, y_2)$ comes from the Pythagorean theorem, and the midpoint is the average of the coordinates:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \qquad M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

The distance from $(1, 2)$ to $(4, 6)$ is $\sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ , and their midpoint is $\left(\tfrac{5}{2}, 4\right)$ .

Reading a Graph

Many GRE items just ask you to read a value off a plotted line. To find $f(3)$ from a graph, locate $x = 3$ on the horizontal axis, move up (or down) to the line, and read the $y$ -value. To solve $f(x) = 0$ , find where the graph crosses the $x$ -axis. Where two graphs intersect, both functions share the same input and output — that point is the solution of the system.

A Reliable Game Plan

Evaluate $f(a)$ ? Substitute $a$ for every $x$ ; for composition, work inside-out.
Domain? Exclude inputs that divide by zero or take an even root of a negative.
Slope? Rise over run, keeping the point order consistent.
Equation of a line? Get the slope, then anchor it with a point via point-slope.
Perpendicular? Negative reciprocal, so the product of slopes is $-1$ .
Distance/midpoint? Distance uses the Pythagorean form; midpoint averages the coordinates.

Common Mistakes to Avoid

Reversing the subtraction order inconsistently in the slope formula, flipping the sign.
Using the plain reciprocal (not the negative reciprocal) for a perpendicular slope.
Forgetting the square root in the distance formula, or subtracting instead of averaging for the midpoint.
Assuming $f(g(x))$ equals $g(f(x))$ — composition is not commutative.

If $f(x) = x^2 - 4x + 7$ , what is $f(5)$ ?

What is the slope of the line through $(-2, 3)$ and $(4, 15)$ ?

Find the distance between $(1, -2)$ and $(7, 6)$ .

A line passes through $(2, 1)$ and $(6, 9)$ . Write its equation in slope-intercept form.

Line $\ell$ has equation $y = \tfrac{3}{2}x - 4$ . Find the equation of the line perpendicular to $\ell$ that passes through $(3, 5)$ .

If $f(x) = 3x - 1$ and $g(x) = x^2 + 2$ , what is $f(g(3))$ ?

Using the plain reciprocal for a perpendicular slope — turning slope $\tfrac{3}{4}$ into $\tfrac{4}{3}$ instead of $-\tfrac{4}{3}$ .

Perpendicular slopes are NEGATIVE reciprocals: flip the fraction AND change the sign so the product of slopes is $-1$ .

Mixing up the subtraction order in the slope formula, e.g. using $y_2 - y_1$ on top but $x_1 - x_2$ on the bottom.

Keep the same order in numerator and denominator. Both start from point 2 (or both from point 1); mixing them flips the sign.

Forgetting the square root in the distance formula, or subtracting coordinates for a midpoint instead of averaging.

Distance takes a square root of the sum of squared gaps; midpoint averages each coordinate. They are different operations.

Evaluating a composition in the wrong order, computing $g(f(3))$ when the question asks for $f(g(3))$ .

Read $f(g(x))$ inside-out: apply the inner function $g$ first, then feed its output into $f$ . Composition is not commutative.

If $f(x) = 2x^2 - x + 3$ , what is $f(-3)$ ?

What is the midpoint of the segment joining $(-4, 7)$ and $(10, -1)$ ?

Which line is perpendicular to the line $2x + 3y = 12$ ?

The distance between $(2, k)$ and $(2, 9)$ is $6$ units, and $k < 9$ . What is the value of $k$ ?

The function $f$ is defined by $f(x) = \dfrac{x + 4}{x - 5}$ . Which value is NOT in the domain of $f$ ?

A line has slope $-\tfrac{1}{2}$ and passes through $(6, 1)$ . What is its $y$ -intercept?

Line $\ell$ passes through the points $(0, 0)$ and $(4, 6)$ .

Quantity A: The slope of line $\ell$

Quantity B: $\dfrac{2}{3}$