Algebra and Functions – Basic 1

(1) Solve the following system of equations for \(x\) and \(y\):
\[ 2x + 3y = 16 \]
\[ x – 4y = -3 \]

• (a) \( x = 2, y = 4 \)
• (b) \( x = 4, y = 2 \)
• (c) \( x = 3, y = 3 \)
• (d) \( x = 5, y = 1 \)

Answer

Correct Answer: (b) \( x = 4, y = 2 \)

To solve the system, use elimination or substitution methods. Here’s a step-by-step solution using the elimination method. 2x + 3y = 16, x - 4y = -3

(2) Find the vertex of the quadratic function \( f(x) = x^2 – 6x + 8 \).

• (a) \( (2, -2) \)
• (b) \( (3, -1) \)
• (c) \( (1, 3) \)
• (d) \( (4, 0) \)

Answer

Correct Answer: (b) \( (3, -1) \)

To find the vertex of a quadratic function, use the vertex formula. The step-by-step solution is: x = -\frac{b}{2a}, f(x) = x^2 - 6x + 8

(3) If \( f(x) = 2x – 1 \) and \( g(x) = x^2 + 4 \), find \( (f \circ g)(2) \) (i.e., \( f(g(2)) \)).

• (a) \( 17 \)
• (b) \( 15 \)
• (c) \( 12 \)
• (d) \( 10 \)

Answer

Correct Answer: (b) \( 15 \)

Find \( (f \circ g)(2) \) by first evaluating \( g(2) \) and then using this value in \( f(x) \). The detailed solution steps are: f(x) = 2x - 1, g(x) = x^2 + 4, f(g(2))

(4) Simplify the expression:
\[ \frac{3x^3}{2x^2} \cdot \frac{4x^2}{6x} \]

• (a) \( \frac{x}{3} \)
• (b) \( 2x \)
• (c) \( \frac{4}{3}x \)
• (d) \( \frac{2}{3}x \)

Answer

Correct Answer: (a) \( \frac{x}{3} \)

To simplify the expression, multiply the fractions and simplify the terms. Here’s the step-by-step solution: \frac{3x^3}{2x^2} \cdot \frac{4x^2}{6x}

(5) If \( f(x) = 2x + 1 \) and \( g(x) = x^2 – 3 \), find \( (g \circ f)(3) \) (i.e., \( g(f(3)) \)).

• (a) \( 14 \)
• (b) \( 11 \)
• (c) \( 10 \)
• (d) \( 7 \)

Answer

Correct Answer: (c) \( 10 \)

Find \( (g \circ f)(3) \) by first evaluating \( f(3) \) and then using this value in \( g(x) \). The detailed solution steps are: f(x) = 2x + 1, g(x) = x^2 - 3, g(f(3))