Algebra and Functions
What the exam tests
Algebra and functions questions test fluent movement between algebraic, graphical and symbolic forms: domains and ranges, composition and inversion, quadratic and rational functions, exponential and logarithmic equations, and inequalities. Expect one direct computation and one multi-step problem in which a domain restriction changes the answer.
Core rules
- Domain: an even-root radicand must be ≥ 0, a denominator must be ≠ 0, and a logarithm argument must be > 0.
- For f∘g, x must lie in D(g) and g(x) must lie in D(f).
- Inverse: solve y=f(x) for x, then swap the variables; D(f⁻¹)=R(f), and the two graphs are mirror images in y=x.
- Quadratic: Δ=b²−4ac; the vertex is at x=−b/(2a); the sign of a gives the concavity.
- For a>0, a≠1: a^u=a^v ⟺ u=v; ln and exp are inverse functions.
- Rational or modulus inequalities are solved with a sign table, not by multiplying blindly.
Worked example
Solve 4^x − 5·2^x + 4 = 0. Write 4^x=(2^x)² and let t=2^x, with t>0. Then t²−5t+4=0, so (t−1)(t−4)=0 and t=1 or t=4. Back-substituting: 2^x=1 gives x=0; 2^x=4 gives x=2. Both values give t>0, so the solution set is {0,2}.
Traps
- Squaring an equation and keeping extraneous roots without checking them in the original equation.
- Writing √(x²)=x instead of |x|.
- Forgetting that an inverse exists only after restricting the domain so that f is injective.
- Reversing the inequality sign only sometimes when dividing by a negative quantity.
- Dropping the t>0 condition in an exponential substitution.