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College Algebra Unit 5: Functions & their properties

A function assigns each input exactly one output. This unit covers how to read, evaluate and combine functions.

Covers: Definition, domain/range, evaluation, algebra of functions, composition

Study guide

A function assigns exactly one output to each input. A graph represents a function if it passes the vertical line test: no vertical line crosses it more than once. In function notation, f(a) means the output for input a; evaluating means substituting carefully, using parentheses for expressions like f(x + h).

The domain is the set of allowed inputs. Exclude values that make a denominator zero (1/(x + 2) needs x ≠ −2) and values that make an even root negative (√(x − 3) needs x ≥ 3). The range is the set of outputs, often read from a graph.

The algebra of functions combines them: (f + g)(x) = f(x) + g(x), and similarly for differences, products and quotients. The domain of a combination is where both functions are defined, minus any zeros of the denominator for quotients.

Composition (f ∘ g)(x) = f(g(x)) applies g first, then f. Evaluate inside out: find g(x), then plug the result into f. In general f(g(x)) ≠ g(f(x)). The domain of f ∘ g includes only x-values in g's domain whose outputs are in f's domain.

The difference quotient [f(x + h) − f(x)]/h measures average rate of change and previews the derivative in calculus. Expand f(x + h), subtract f(x), and simplify before dividing by h.

A function is increasing on an interval if its outputs rise as the inputs rise, and decreasing if they fall. The average rate of change from a to b is [f(b) − f(a)]/(b − a), the slope of the secant line. To evaluate f(g(2)) from a table or graph, first read g(2), then find f at that output.

For a sum such as √(x − 1) + √(5 − x), each radicand must be at least 0. The domain is the overlap of the separate domains, here [1, 5].

Key ideas

• Vertical line test: a graph is a function if no vertical line crosses it twice.
• Domain is the set of allowed inputs. Exclude values that divide by zero or take even roots of negatives.
• Range is the set of outputs the function actually produces.
• Combine functions with +, −, ×, ÷; the domain is where both are defined (and the divisor isn't zero).
• Composition (f ∘ g)(x) = f(g(x)) means apply g first, then f.

Formulas and rules

Common mistakes

• Computing f(g(x)) in the wrong order.
• Writing f(x + h) = f(x) + h.

Words to know

Vertical line test

A graph is a function if no vertical line hits it more than once.

Domain of a square root function

The radicand must be ≥ 0: √(x − 3) needs x ≥ 3.

Domain of a rational function

All real numbers except where the denominator is 0.

Composition (f ∘ g)(x)

f(g(x)): apply g first, then f.

Evaluating a composition

Work inside out: find g(x), then plug the result into f.

Difference quotient

[f(x + h) − f(x)] / h

Function

A relation giving exactly one output for each input.

Domain

The set of allowed inputs.

Range

The set of outputs.

Sum of functions

(f + g)(x) = f(x) + g(x)

Quotient of functions

(f/g)(x) = f(x)/g(x), with g(x) ≠ 0.

Domain of f ∘ g

x in g's domain with g(x) in f's domain.

Sample practice questions

23 more questions like these, from level 1 to 5, are in the practice set.