College Algebra on ALEKS

College Algebra questions on ALEKS test formal function theory: polynomial roots, synthetic division, rational function behavior (asymptotes), function transformations, and function inverses.

Core Tested Concepts

Function Operations & InversesFunction composition (f ∘ g)(x) = f(g(x)), finding inverse functions f⁻¹(x) by swapping x and y, verifying one-to-one via horizontal line test.

Rational Zeros & Remainder TheoremRational Zero Theorem (factors of p over factors of q), evaluating polynomial remainder P(c) when dividing by (x - c).

Rational Functions & AsymptotesFinding vertical asymptotes (zeros of denominator), horizontal asymptotes (degree comparison), and holes (removable discontinuities).

Function TransformationsVertical shifts f(x) + c, horizontal shifts f(x - c), reflections across x-axis (-f(x)) and y-axis (f(-x)), and vertical stretching.

High-Yield Cheat Sheet

Essential College Algebra Theorems

Remainder Theorem

P(c) = Remainder

Composition

(f ∘ g)(x) = f(g(x))

Inverse Property

f(f⁻¹(x)) = x

Practice Drills

5 Authentic College Algebra Practice Questions

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1Practice Problem

Given f(x) = 3x - 5 and g(x) = 2x² + 1, find the value of (g ∘ f)(2).

Formula: (g ∘ f)(x) = g(f(x))

2Practice Problem

Find the horizontal asymptote of the rational function: R(x) = (6x² - 5) / (2x² + 7x + 1)

Formula: Equal degrees: y = a_n / b_n

3Practice Problem

According to the Remainder Theorem, what is the remainder when P(x) = 2x³ - 5x² + 4x - 7 is divided by (x - 2)?

Formula: Remainder = P(c)

4Practice Problem

If f(x) = (5x + 4) / 3, find the inverse function value f⁻¹(8).

Formula: f⁻¹(y) = x <=> f(x) = y

5Practice Problem

What is the domain of the rational function f(x) = 7 / (x² - 9)?

Formula: Denominator ≠ 0

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