- Evaluate expressions like f(g(x)) and recognize graph transformations
- Use the link between a polynomial's zeros and its factors
- Solve radical and rational equations and reject extraneous solutions
The second half of Advanced Math is questions that look scary but follow a few rules: f(g(x)), x³ − 4x² + x + 6, x^(3/2), √(x + 7) = x − 5. Most of them take 3–4 lines — if you know the rule. After each topic we will also show a fast way to check your answer.
- 1Rewrite what is given
Copy f(x), g(x) or the polynomial onto scratch paper so it is in front of you.
- 2Note restrictions
A denominator cannot be zero and an expression under a square root cannot be negative — write this down before solving.
- 3Inside out
For compositions and transformations, work out the inside of the brackets first.
- 4Check
Check every root in the original equation; plug in a number to test equivalence.
Function notation and transformations
f(3) means “put 3 in place of x”. f(g(x)) is worked from the inside out: first g, then f. For example, if f(x) = 2x + 1 and g(x) = x², then f(g(2)) = f(4) = 9, but g(f(2)) = g(5) = 25. Order matters!
- hthe graph shifts h units right (left if h < 0)
- kthe graph shifts k units up (down if k < 0)
- avertical stretch; if a < 0 the graph is reflected over the x-axis
f(−x) reflects the graph over the y-axis.
The graph of y = f(x) passes through (2, 5). Which point lies on the graph of y = f(x − 3) + 1?
Show solutionHide solution
+1 → 1 unit up: 5 + 1 = 6.
Answer: (5, 6).
Check: at x = 5, f(5 − 3) + 1 = f(2) + 1 = 5 + 1 = 6.
Polynomials: zeros, factors and remainders
- p(x)a polynomial
- aa zero of the polynomial: the graph crosses the x-axis at x = a
Remainder theorem: when p(x) is divided by (x − a), the remainder equals p(a).
p(x) = x³ − 4x² + x + 6. Find p(2) and all the zeros of the polynomial.
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Divide: x³ − 4x² + x + 6 = (x − 2)(x² − 2x − 3).
x² − 2x − 3 = (x − 3)(x + 1).
So p(x) = (x − 2)(x − 3)(x + 1), and the zeros are −1, 2, 3.
Check: the zeros add up to 4, the opposite of the x² coefficient.
Rational exponents, radical and rational equations
- nthe index of the root (the denominator)
- mthe power (the numerator)
For example: 8^(2/3) = (∛8)² = 2² = 4.
a) Solve √(x + 7) = x − 5.
b) Solve x/(x − 2) = 2/(x − 2) + 3.
Show solutionHide solution
Check: x = 9: √16 = 4 = 9 − 5 ✓; x = 2: √9 = 3, but 2 − 5 = −3 ✗.
Answer: only x = 9 (x = 2 is extraneous).
b) Multiply both sides by (x − 2): x = 2 + 3(x − 2) ⇒ x = 3x − 4 ⇒ x = 2.
But x = 2 makes the denominator zero! So there is no solution.
Key points
- f(g(x)) is evaluated from the inside; f(g(x)) and g(f(x)) are usually different.
- f(x − h) + k: h units right, k units up.
- p(a) = 0 ⇔ (x − a) is a factor; the remainder on division by (x − a) is p(a).
- x^(m/n) = ⁿ√(xᵐ).
- In radical and rational equations, check every root in the original equation.
Check yourself
10 questions. Every correct answer earns XP.