Algebra and Functions
The Algebra and Functions domain of the CLEP College Mathematics exam asks you to manipulate expressions, solve equations and inequalities, work fluently with function notation, and model real situations with linear, quadratic, and exponential functions. Questions are short but reward precision: most wrong answers come from a single sign slip or a misread domain restriction rather than from deep theory.
Function notation and domains
Write f(x) = (x + 3)/(x² − 4). Evaluating f(2) means substituting 2 for every x, so f(2) = 5/0, which is undefined - that single value is excluded from the domain. In general, the domain of a rational function excludes zeros of the denominator, and the domain of an even root excludes values making the radicand negative. Range questions ask which outputs are possible, so solve y = f(x) for x and check when that solution is real.
Composition and inverses
(f∘g)(x) = f(g(x)) - apply the inner function first. Composition is not commutative, so (f∘g)(2) and (g∘f)(2) are usually different numbers. An inverse f⁻¹ exists only where f is one-to-one (passes the horizontal line test); to find it, swap x and y and solve for y.
Exponentials and logarithms
y = a·b^x grows when b > 1 and decays when 0 < b < 1. A 5% annual increase is a multiplier of 1.05, not an addition of 5. The definition log_b(x) = y means b^y = x, and the product rule log(AB) = log A + log B turns multiplicative growth into additive statements.
Worked example
Let f(x) = (x + 3)/(x² − 4) and g(x) = 2x − 1. Find the domain of f and the value of (f∘g)(2). Domain: x² − 4 = 0 at x = 2 and x = −2, so the domain is all reals except ±2. (f∘g)(2) = f(g(2)) = f(2·2 − 1) = f(3) = (3 + 3)/(9 − 4) = 6/5. Reversing the order gives g(f(2)), but f(2) is undefined - a reminder that order matters.
Traps
- Squaring both sides of an equation can create extraneous roots; always check candidates in the original equation.
- Multiplying or dividing an inequality by a negative number reverses the inequality sign.
- Cancelling a common factor such as (x − 2) simplifies the expression but does not put x = 2 back into the domain.
- Growth stated as a percent per year is exponential, not linear; two years at 10% is a factor of 1.21, not 1.20.