Functions and Transformations: math test, Grade 11 – download the PDF

1 Notation and domains / 3 pts

Calculator not allowed.

1. If \( f(x) = 2x^2 - x + 3 \), compute \( f(-2) \).
2. Find the domain of \( h(x) = \sqrt{2x - 7} \).
3. Find the domain of \( k(x) = \dfrac{x + 1}{x^2 - 16} \).

2 Transformations / 4 pts

Let \( g(x) = -3|x + 2| + 5 \).

1. Describe how the graph of \( y = |x| \) is transformed into the graph of \( g \).
2. Give the vertex of \( g \).
3. Give the range of \( g \).
4. Compute \( g(1) \).

3 Combining functions / 3 pts

Let \( f(x) = x^2 + 1 \) and \( g(x) = 2x - 3 \).

1. Find \( (f+g)(x) \).
2. Compute \( (fg)(2) \).
3. Give the domain of \( \dfrac{g}{f} \) and justify.

4 Composition / 4 pts

Let \( f(x) = 3x + 2 \) and \( g(x) = x^2 - 1 \).

1. Find \( f(g(x)) \).
2. Find \( g(f(x)) \).
3. Compute \( f(g(-2)) \).
4. Solve \( f(g(x)) = 47 \).

5 Inverse functions / 4 pts

Let \( f(x) = (x - 2)^3 + 4 \).

1. Find \( f^{-1}(x) \).
2. Compute \( f^{-1}(-4) \) and check it with \( f \).
3. Show that \( f(f^{-1}(x)) = x \).

6 Shipping cost / 2 pts

A shop charges \$6 to ship a package of at most 2 pounds, and \$2.50 more per additional pound (prorated) beyond 2 pounds: \( S(w) = 6 \) if \( 0 \lt w \le 2 \), and \( S(w) = 6 + 2.5(w - 2) \) if \( w \gt 2 \). Compute \( S(1.2) \) and \( S(6) \).