Chapter Review: Worked Practice Problems

This review pulls the whole chapter together — functions, inequalities, matrices, complex numbers, quadratics and exponent rules — through worked problems of the kind an exam actually asks.

Example 1 — The vertical line test

A relation is given by this table. Does it represent a function?

x: −2, −1, 0, 1, 2

y: 5, 2, 1, 2, 5

Each x value appears once and gives exactly one y.

Any vertical line meets the curve at a single point.

⟹ yes, it is a function

Repeated y values are fine. It is a repeated x with two different outputs that would break the rule.

Example 2 — Graphing an inequality

For

the boundary is drawn solid because

includes it. Testing

gives

, which is true — so the region containing the origin is the one to shade.

Example 3 — Multiplying matrices

Multiply by .

Entry (1,1):

Entry (1,2):

Entry (2,1):

Entry (2,2):

Each entry is a row of the first matrix paired against a column of the second.

Example 4 — Inverting a diagonal matrix

Find the inverse of .

Swap the main diagonal and flip the other signs:

, then divide by 6:

For a diagonal matrix the inverse is simply the reciprocal of each diagonal entry.

Example 5 — Complex numbers

Conjugate product:

⟹ 5, a real number

Root of a negative:

⟹ 11i

Example 6 — Solving with the general formula

Solve , so .

⟹ x = −6 ± 3√5

Example 7 — Reading the discriminant

For

⟹ Δ > 0, two distinct real roots

For

⟹ Δ < 0, two complex conjugate roots

Example 8 — Exponent rules

Handle the coefficients first, then each letter separately.

Coefficients:

Powers of a:

Powers of b:

⟹ −8a⁵b²

: coefficients give 6,

⟹ 6x³ / y³

Summary

A relation is a function when every x gives exactly one y.
Use a solid boundary for ≥ or ≤, and test a point to pick the region.
Matrix entries come from row-by-column products; the inverse needs det ≠ 0.
A conjugate product is real, and √(−121) = 11i.
The discriminant classifies the roots before you solve for them.