📝 Chapter 1: Introduction to Functions

Which of the following is NOT a function?

For \( f(x) = -2x^2 + 5x - 3 \), evaluate \( f(-1) \).

State the domain of \( g(x) = \dfrac{2x + 1}{x^2 - 9} \).

Find the inverse of \( f(x) = 4x - 9 \). Show all algebraic steps.

Determine whether \( f(x) = 5x^3 - 2x \) is even, odd, or neither.

Sketch \( y = (x + 2)^2 - 5 \) using transformations of \( y = x^2 \). State vertex and direction of opening.

Given \( f(x) = x^2 + 3x \) and \( g(x) = 2x - 1 \), determine: (a) \( f(g(2)) \); (b) \( g(f(2)) \); (c) explain why composition is generally not commutative.

For \( f(x) = -x^2 + 4 \): (a) state the domain and range; (b) find \( f^{-1}(x) \) restricted so that the inverse is a function; (c) state the domain and range of \( f^{-1} \).

For \( f(x) = 3x - 7 \), find \( \dfrac{f(2 + h) - f(2)}{h} \) and explain what this expression represents.

A function \( f \) is even AND odd. Prove that \( f(x) = 0 \) for every \( x \) in its domain.

Explain in your own words the difference between a relation and a function. Use one example of each.

Describe the geometric relationship between the graph of \( f(x) \) and \( f^{-1}(x) \). Mention the line of reflection and one specific example.

Describe each transformation in \( g(x) = -3 f(2(x - 1)) + 5 \), in the order they should be applied. Include effects on key points such as a vertex.

For \( f(x) = \sqrt{x + 3} - 2 \): identify the parent function, list the transformations in order, state the domain and range, and explain how restrictions arise from the radical.

A swimming pool's water depth (m) is modelled by \( d(t) = 0.5t + 1 \), where \( t \) is the time (hours) after the pool starts filling. Find: (a) the depth at \( t = 5 \) h; (b) the time when depth is 4 m; (c) state the domain and range that make sense for this real-world context if the pool is full at depth 5 m.

A rectangular dog run uses 30 m of fencing. The area is \( A(w) = w(15 - w) \), where \( w \) is the width. Find: (a) the area when \( w = 5 \) m; (b) the value of \( w \) that maximizes the area; (c) the maximum area.

A taxi charges \$3.50 plus \$1.25 per km. Let \( C(d) = 3.50 + 1.25d \). (a) Find the inverse \( C^{-1}(x) \); (b) interpret what \( C^{-1}(x) \) represents in this context; (c) Aretha was charged \$16.00 — how far did she travel?

A ball's height (m) above the ground is \( h(t) = -5(t - 2)^2 + 20 \), where \( t \) is in seconds. Use transformations of the parent \( y = t^2 \) to identify the maximum height and the time it occurs. Justify with reference to the vertex form.