Functions and Transformations: math practice, Grade 11

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Math practice Grade 11 : Functions and Transformations — Zyro the alien explorer of Planète Maths

22 problems graded by difficulty (★ easy, ★★ medium, ★★★ hard). Try first, without looking at the solutions!

Sommaire

1 Evaluating a function ★★★

Let $ f(x) = 3x^2 - 2x + 5 $. Compute $ f(-2) $, $ f(0) $ and $ f(3) $.

2 Is it a function? ★★★

Consider the two relations $ A = \{(-3, 4), (-1, 2), (0, 2), (2, -5), (4, 1)\} $ and $ B = \{(1, 3), (2, 5), (1, -3)\} $.

Which one is a function? Justify.
For the function, give the domain and the range.

3 Finding domains ★★★

Find the domain of each function.

$ f(x) = \sqrt{x - 4} $
$ g(x) = \dfrac{5}{x + 3} $
$ h(x) = \sqrt{3x + 12} $

4 Name the shift ★★★

Name the parent function and describe the transformation for each graph.

$ g(x) = (x + 4)^2 - 7 $
$ h(x) = -|x| + 2 $
$ k(x) = \sqrt{x - 5} $

5 Sum, difference, product ★★★

Let $ f(x) = 2x + 3 $ and $ g(x) = x - 6 $. Find $ (f+g)(x) $, $ (f-g)(x) $, $ (fg)(x) $, then compute $ (f+g)(4) $.

6 First composition ★★★

Let $ f(x) = x + 5 $ and $ g(x) = 2x $.

Compute $ f(g(3)) $ and $ g(f(3)) $.
Find $ f(g(x)) $ and $ g(f(x)) $. Are they equal?

7 A first inverse ★★★

Find the inverse of $ f(x) = 4x - 7 $, then check your answer with the value $ f^{-1}(5) $.

8 Piecewise evaluation ★★★

A function is defined by \[ f(x) = \begin{cases} x + 4 & \text{if } x \lt -1 \\ x^2 & \text{if } -1 \le x \le 2 \\ 6 - x & \text{if } x \gt 2 \end{cases} \] Compute $ f(-3) $, $ f(-1) $, $ f(0) $, $ f(2) $ and $ f(5) $.

9 Build the equation ★★★

The graph of $ y = x^2 $ is reflected over the x-axis, shifted 3 units right, then 5 units up.

Write the equation of the new function $ g $.
Give its vertex and compute $ g(1) $.

10 Taxi fare ★★★

A taxi charges a base fee of \$3 plus \$2.50 per mile. The fare for a ride of $ m $ miles is $ C(m) = 3 + 2.5m $.

Give the domain and the range of $ C $ in this context.
Find $ C(12) $.
Find the length of a ride that costs \$48.
Find the inverse function and explain what it computes.

11 Composition with a quadratic ★★★

Let $ f(x) = x^2 - 3 $ and $ g(x) = 2x + 1 $. Find $ f(g(x)) $ and $ g(f(x)) $, then compute $ f(g(2)) $ and $ g(f(2)) $.

12 A quotient with a hole ★★★

Let $ f(x) = x^2 - 9 $ and $ g(x) = x - 3 $.

Simplify $ \left(\dfrac{f}{g}\right)(x) $ and give its domain.
Give the domain of $ \left(\dfrac{g}{f}\right)(x) $.

13 Inverse of a rational function ★★★

Find the inverse of $ f(x) = \dfrac{3}{x - 1} + 2 $ and verify it using the value $ f(4) $.

14 Absolute value equation and inequality ★★★

Solve.

$ |2x - 5| = 9 $
$ |x - 3| \lt 4 $ (describe the answer on a number line and in interval notation).

15 Domain and range from a transformation ★★★

Let $ g(x) = -2\sqrt{x + 3} + 1 $.

Give the domain and range of $ g $.
Compute $ g(1) $ and solve $ g(x) = -5 $.
Describe the transformation of $ \sqrt{x} $ and compare with the graph.

16 Inverse with a restricted domain ★★★

Let $ f(x) = (x - 3)^2 + 1 $ with the domain restricted to $ x \ge 3 $.

Explain why the restriction is needed.
Find $ f^{-1}(x) $, with its domain and range.
Check with $ f(5) $.

17 Does the order matter? ★★★

Let $ f(x) = \sqrt{x} $ and $ g(x) = x - 4 $.

Find $ f(g(x)) $ and $ g(f(x)) $.
Find the domain of each composition.
Are the two compositions equal? Justify with a value of $ x $.

18 Parking fees ★★★

A garage charges \$4 for the first 2 hours. After that it adds \$3 for each additional hour (prorated), and the total is capped at \$25. With $ t $ in hours, \[ P(t) = \begin{cases} 4 & \text{if } 0 \lt t \le 2 \\ 4 + 3(t - 2) & \text{if } 2 \lt t \le 9 \\ 25 & \text{if } t \gt 9 \end{cases} \]

Compute $ P(1.5) $, $ P(5) $ and $ P(12) $.
For which time is the fee \$19?
Explain why the cap starts at 9 hours, and sketch the graph.

19 Absolute value as a piecewise function ★★★

Let $ h(x) = 2|x - 1| - 3 $.

Write $ h $ as a piecewise function without absolute values.
Find the zeros of $ h $.
Give the vertex and the range.

20 Discount or coupon first? ★★★

A store offers a 20% discount, and you also have a \$5 coupon. Let $ f(x) = 0.8x $ (the discount) and $ g(x) = x - 5 $ (the coupon), where $ x $ is the original price in dollars.

For a \$60 jacket, compute $ g(f(60)) $ and $ f(g(60)) $.
Find $ g(f(x)) $ and $ f(g(x)) $. Which order is cheaper, and by how much?

21 Transforming a table of values ★★★

The function $ f $ has these values:

$ x $	$ -2 $	$ 0 $	$ 1 $	$ 4 $
$ f(x) $	$ 1 $	$ 3 $	$ -2 $	$ 0 $

Give four points of the graph of $ g(x) = 2f(x+1) - 3 $.
Give four points of $ h(x) = f(2x) $.

22 Proving two functions are inverses ★★★

Let $ f(x) = \dfrac{2x + 1}{x - 3} $ and $ g(x) = \dfrac{3x + 1}{x - 2} $. Show that $ f(g(x)) = x $. What does this tell you, and what would you check to be fully sure?