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Problem-solving Pack

Ecolint Campus des NationsMathematics
Year 11 · 11.3 Functions

Problem-solving Pack

Name: _________________________________
Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12
Evaluate & solve. Let f(x)=2x+3f(x) = 2x + 3.

(a) Find f(5)f(5).
(b) Find f(−1)f(-1).
(c) Find the value of xx for which f(x)=11f(x) = 11.

Working space

2Problem 2 of 12
Natural domains. State the largest natural domain of each function over R\mathbb{R}.

(a) f(x)=1x−3f(x) = \dfrac{1}{x - 3}
(b) g(x)=x−1g(x) = \sqrt{x - 1}
(c) h(x)=14−xh(x) = \dfrac{1}{\sqrt{4 - x}}

Working space

3Problem 3 of 12
Vertical line test. State whether each relation defines yy as a function of xx. Justify.

(a) y=x2y = x^2
(b) x=y2x = y^2
(c) x2+y2=9x^2 + y^2 = 9
(d) y=∣x∣y = |x|

Working space

4Problem 4 of 12
Composition & inverse [EXT]. Let f(x)=2x+5f(x) = 2x + 5 and g(x)=x2−3g(x) = x^2 - 3.

(a) Find f(g(3))f(g(3)).
(b) Find a simplified expression for (f∘g)(x)(f \circ g)(x).
(c) Find f−1(x)f^{-1}(x) and state its domain.

Working space

5Problem 5 of 12
Domain of a composition [EXT]. Let f(x)=xf(x) = \sqrt{x} and g(x)=5−x2g(x) = 5 - x^2.

(a) Find (f∘g)(x)(f \circ g)(x).
(b) State its largest natural domain.
(c) State its range.

Working space

6Problem 6 of 12
Piecewise function. Let f(x)={2x+1x<0x20≤x≤39x>3f(x) = \begin{cases} 2x + 1 & x < 0 \\ x^2 & 0 \leq x \leq 3 \\ 9 & x > 3 \end{cases}.

(a) Find f(−2)f(-2), f(0)f(0), f(2)f(2), f(5)f(5).
(b) Sketch ff for −3≤x≤5-3 \leq x \leq 5.
(c) State the range of ff.

Working space

7Problem 7 of 12
Absolute value [EXT]. Solve and sketch.

(a) Solve ∣2x−3∣=7|2x - 3| = 7.
(b) Solve ∣x−1∣<4|x - 1| < 4 and write the answer in interval notation.
(c) Sketch y=∣x−2∣−1y = |x - 2| - 1, marking xx- and yy-intercepts.

Working space

8Problem 8 of 12
Modelling with a linear function. A car rental charges a fixed daily fee FF (CHF) plus a charge cc (CHF) per km. A driver pays CHF 95 for 200 km in a day, and CHF 125 for 350 km in a day.

(a) Write a function C(d)=cd+FC(d) = cd + F for the cost in terms of the distance dd (km).
(b) Find cc and FF.
(c) Predict the cost of a 500 km day.

Working space

9Problem 9 of 12
Finding f from data — quadratic. A quadratic function ff passes through (0,3)(0, 3), (1,6)(1, 6), and (2,13)(2, 13).

(a) Assume f(x)=ax2+bx+cf(x) = ax^2 + bx + c. Set up three equations.
(b) Solve to find aa, bb, cc.
(c) State f(−1)f(-1).

Working space

10Problem 10 of 12
Range of a quadratic. Let f(x)=x2−4x+7f(x) = x^2 - 4x + 7.

(a) Express f(x)f(x) in vertex form.
(b) State the minimum value and where it occurs.
(c) State the range of ff.

Working space

11Problem 11 of 12
Inverse from a graph [EXT]. The graph of ff is a straight line through (−2,1)(-2, 1) and (4,4)(4, 4).

(a) Find f(x)f(x).
(b) Find f−1(x)f^{-1}(x).
(c) On a single set of axes, sketch y=f(x)y = f(x), y=f−1(x)y = f^{-1}(x) and y=xy = x. State the symmetry.

Working space

12Problem 12 of 12
Mini-investigation — natural domain. Three students propose different formulas for the same function:

- Anya: f(x)=x2−1x−1f(x) = \dfrac{x^2 - 1}{x - 1}
- Bao: f(x)=x+1f(x) = x + 1
- Cara: f(x)=x+1f(x) = x + 1 for x≠1x \neq 1

(a) Compute f(2)f(2) using each formula.
(b) State the natural domain of Anya's formula.
(c) Are Anya's and Bao's functions equal? Justify.
(d) Whose formula is most precise? Justify.

Working space