7.2 Relations, Functions, Equations & Inequalities

7.2 Relations, Functions, Equations & Inequalities

Quick Answer: A function maps each input to exactly one output; test graphs with the vertical line test. Inequality rules mirror equation rules, except multiplying or dividing by a negative number flips the inequality sign.

Relations and Functions

A relation is any set of ordered pairs (input, output). A function is a relation where each input (x-value in the domain) pairs with exactly one output (y-value in the range).

• Domain: the set of all valid inputs (x-values).
• Range: the set of all outputs produced (y-values).

Vertical Line Test

On a graph, the relation is a function if and only if no vertical line intersects the graph more than once. A circle fails the test (a vertical line hits it twice); a parabola passes (any vertical line hits it once at most).

Function Notation

f(x) reads "f of x" and means "the output of f at input x." For f(x) = 3x^2 - 2x + 1:

• f(2) = 3(2)^2 - 2(2) + 1 = 12 - 4 + 1 = 9.
• f(a) = 3a^2 - 2a + 1 (treat a like any other input).

The GRE may ask you to evaluate composite functions: f(g(x)) means "evaluate g first, then put the result into f." Example with f(x) = 2x + 1 and g(x) = x^2: f(g(3)) = f(9) = 2(9) + 1 = 19.

Equations vs. Inequalities

An equation asserts equality using =. An inequality describes a range using <, >, <=, or >= rather than a single value.

Inequality Symbols

Inequality Operations - The Critical Flip Rule

Inequality rules mirror equation rules except for one critical case:

Worked Example 4: Solve -3x + 5 <= 17.

• Subtract 5: -3x <= 12.
• Divide by -3 and FLIP: x >= -4.

Students lose points by forgetting to flip. Always check: did I multiply or divide by a negative? If yes, flip the sign.

Solving Linear Inequalities

Solve 4(x - 2) > 2x + 6.

• Distribute: 4x - 8 > 2x + 6.
• Subtract 2x: 2x - 8 > 6.
• Add 8: 2x > 14.
• Divide by 2 (positive, no flip): x > 7.

The solution is the open interval (7, infinity). On the GRE, you may be asked which values satisfy this - anything greater than 7.

Compound Inequalities

Two inequalities joined by "and" or "or."

• "And" (conjunction): both must hold. The solution is the overlap. Example: x > 2 AND x < 5 gives 2 < x < 5.
• "Or" (disjunction): either holds. The solution is the union. Example: x < -1 OR x > 3 gives (-infinity, -1) union (3, infinity).

Functions on the GRE - Common Forms

The GRE tests these function forms often:

• Linear: f(x) = mx + b. Domain all real numbers; range all reals.
• Quadratic: f(x) = ax^2 + bx + c. Parabola; domain all reals; range depends on direction.
• Absolute value: f(x) = |x|. V-shape; range [0, infinity).
• Square root: f(x) = sqrt(x). Domain x >= 0; range y >= 0.
• Rational: f(x) = 1/x. Domain excludes 0; range excludes 0.

Domain Restrictions to Recognize

1. Denominators can't be zero: f(x) = 1/(x - 2) has domain x != 2.
2. Even roots need non-negative radicands: f(x) = sqrt(x - 4) has domain x >= 4.
3. Logarithms need positive arguments: not on the GRE General Test (no trig/calc), but if encountered, domain x > 0.

Worked Example 5: Find the domain of f(x) = sqrt(2x - 6).

• Need 2x - 6 >= 0, so 2x >= 6 and x >= 3.
• Domain: [3, infinity).

Range - How to Find It

Finding the range takes more work than the domain because you must analyze the function's behavior.

• Linear f(x) = mx + b (m != 0): range is all reals.
• Quadratic f(x) = ax^2 + bx + c: parabola. If a > 0, range is [vertex y, infinity). If a < 0, range is (-infinity, vertex y].
• Square root f(x) = sqrt(x): range is [0, infinity).
• Absolute value f(x) = |x|: range is [0, infinity).
• Rational f(x) = 1/x: range is all reals except 0.

The vertex of a parabola ax^2 + bx + c sits at x = -b / (2a). Plug that x back in to get the y-coordinate.

Worked Example 6: Find the range of f(x) = -x^2 + 4x - 1.

• a = -1 < 0, so parabola opens downward and the vertex is a maximum.
• x-coordinate of vertex: x = -b / (2a) = -4 / (2 * -1) = 2.
• y-coordinate: f(2) = -(2)^2 + 4(2) - 1 = -4 + 8 - 1 = 3.
• Range: (-infinity, 3].

Quick Strategic Points

• For function graphs, scan for vertical line test failures.
• For composite functions, work inside-out: g first, then f.
• For inequalities, always check the flip.
• For domain questions, scan denominators and even roots.
• For range questions on quadratics, find the vertex y-coordinate using x = -b / (2a).
• For QC function comparisons, test boundary points and a value inside the domain.