Advanced: Advanced Math | SAT Math

What You'll Learn

Analyze composition of functions, inverse functions, and piecewise functions.

Perform long division and synthetic division of polynomials.

Solve systems of nonlinear equations and equations with multiple variables.

Apply the remainder theorem, factor theorem, and rational root theorem.

•Function Composition: (f ∘ g)(x) = f(g(x)) - apply functions in order
•Inverse Functions: f⁻¹(f(x)) = x and finding inverses by swapping x and y
•Remainder Theorem: When P(x) is divided by (x - a), remainder is P(a)
•Rational Root Theorem: Possible rational roots are ±(factors of constant)/(factors of leading coefficient)
•End Behavior: Determine how polynomial graphs behave as x approaches ±∞

If f(x) = 2x + 1 and g(x) = x² - 3, find (f ∘ g)(2).

Solution:

(f ∘ g)(2) = f(g(2))

First, find g(2): g(2) = 2² - 3 = 4 - 3 = 1

Then, find f(1): f(1) = 2(1) + 1 = 3

Therefore, (f ∘ g)(2) = 3

✓When composing functions, work from the inside out - evaluate the inner function first
✓Use synthetic division when the divisor is in the form (x - a) for efficiency
✓To find inverse functions, ensure the original function is one-to-one
✓List all possible rational roots systematically before testing them

Ready to test your skills? Choose how you want to study: