Chapter Review: Worked Practice Problems

Eight worked exam-style problems covering functions, graphing inequalities, matrix multiplication and inverses, complex numbers, the quadratic formula, the discriminant and exponent rules.

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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
(0, 0) solid line: boundary included
For y \geq |x| - 4 the boundary is drawn solid because \geq includes it. Testing (0,0) gives 0 \geq -4 , which is true — so the region containing the origin is the one to shade.
Example 3 — Multiplying matrices

Multiply \begin{pmatrix} 2 & 1 \\ 3 & 0 \end{pmatrix} by \begin{pmatrix} 1 & 4 \\ 2 & 3 \end{pmatrix} .

Entry (1,1): 2(1) + 1(2) = 4
Entry (1,2): 2(4) + 1(3) = 11
Entry (2,1): 3(1) + 0(2) = 3
Entry (2,2): 3(4) + 0(3) = 12
\begin{pmatrix} 4 & 11 \\ 3 & 12 \end{pmatrix}

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 \begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix} .

\det = (3)(2) - (0)(0) = 6
Swap the main diagonal and flip the other signs:
\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} , then divide by 6:
A^{-1} = \begin{pmatrix} 1/3 & 0 \\ 0 & 1/2 \end{pmatrix}

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

Example 5 — Complex numbers
Conjugate product: (2 + i)(2 - i) = 4 - i^2 = 4 + 1
⟹ 5, a real number
Root of a negative: \sqrt{-121} = \sqrt{121} \times \sqrt{-1}
⟹ 11i
Example 6 — Solving with the general formula

Solve x^2 + 12x - 9 = 0 , so a = 1, b = 12, c = -9 .

\Delta = 144 - 4(1)(-9) = 144 + 36 = 180
\sqrt{180} = 6\sqrt{5}
x = \frac{-12 \pm 6\sqrt{5}}{2}
⟹ x = −6 ± 3√5
Example 7 — Reading the discriminant
For 3x^2 + 8x + 2 = 0 : \Delta = 64 - 24 = 40
⟹ Δ > 0, two distinct real roots
For 2x^2 - 6x + 9 = 0 : \Delta = 36 - 72 = -36
⟹ Δ < 0, two complex conjugate roots
Example 8 — Exponent rules

Handle the coefficients first, then each letter separately.

(2a^3b^{-2})(-4a^2b^4)
Coefficients: 2 \times (-4) = -8
Powers of a: a^3 \times a^2 = a^5
Powers of b: b^{-2} \times b^4 = b^2
⟹ −8a⁵b²
\dfrac{12x^4y^2}{2xy^5} : coefficients give 6, x^4 \div x = x^3 , y^2 \div y^5 = y^{-3}
⟹ 6x³ / y³
Summary
  1. A relation is a function when every x gives exactly one y.
  2. Use a solid boundary for ≥ or ≤, and test a point to pick the region.
  3. Matrix entries come from row-by-column products; the inverse needs det ≠ 0.
  4. A conjugate product is real, and √(−121) = 11i.
  5. The discriminant classifies the roots before you solve for them.
231. Arithmetic Sequences and Series
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