Problem-solving
11.3 Functions
Show all working. Partial marks are given for method.
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1**Evaluate & solve.** Let $f(x) = 2x + 3$. (a) Find $f(5)$. (b) Find $f(-1)$. (c) Find the value of $x$ for which $f(x) = 11$.
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2**Natural domains.** State the largest natural domain of each function over $\mathbb{R}$. (a) $f(x) = \dfrac{1}{x - 3}$ (b) $g(x) = \sqrt{x - 1}$ (c) $h(x) = \dfrac{1}{\sqrt{4 - x}}$
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3**Vertical line test.** State whether each relation defines $y$ as a function of $x$. Justify. (a) $y = x^2$ (b) $x = y^2$ (c) $x^2 + y^2 = 9$ (d) $y = |x|$
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4**Composition & inverse [EXT].** Let $f(x) = 2x + 5$ and $g(x) = x^2 - 3$. (a) Find $f(g(3))$. (b) Find a simplified expression for $(f \circ g)(x)$. (c) Find $f^{-1}(x)$ and state its domain.
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5**Domain of a composition [EXT].** Let $f(x) = \sqrt{x}$ and $g(x) = 5 - x^2$. (a) Find $(f \circ g)(x)$. (b) State its largest natural domain. (c) State its range.
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6**Piecewise function.** Let $f(x) = \begin{cases} 2x + 1 & x < 0 \\ x^2 & 0 \leq x \leq 3 \\ 9 & x > 3 \end{cases}$. (a) Find $f(-2)$, $f(0)$, $f(2)$, $f(5)$. (b) Sketch $f$ for $-3 \leq x \leq 5$. (c) State the range of $f$.
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7**Absolute value [EXT].** Solve and sketch. (a) Solve $|2x - 3| = 7$. (b) Solve $|x - 1| < 4$ and write the answer in interval notation. (c) Sketch $y = |x - 2| - 1$, marking $x$- and $y$-intercepts.
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8**Modelling with a linear function.** A car rental charges a fixed daily fee $F$ (CHF) plus a charge $c$ (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 $C(d) = cd + F$ for the cost in terms of the distance $d$ (km). (b) Find $c$ and $F$. (c) Predict the cost of a 500 km day.
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9**Finding f from data — quadratic.** A quadratic function $f$ passes through $(0, 3)$, $(1, 6)$, and $(2, 13)$. (a) Assume $f(x) = ax^2 + bx + c$. Set up three equations. (b) Solve to find $a$, $b$, $c$. (c) State $f(-1)$.
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10**Range of a quadratic.** Let $f(x) = x^2 - 4x + 7$. (a) Express $f(x)$ in vertex form. (b) State the minimum value and where it occurs. (c) State the range of $f$.
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11**Inverse from a graph [EXT].** The graph of $f$ is a straight line through $(-2, 1)$ and $(4, 4)$. (a) Find $f(x)$. (b) Find $f^{-1}(x)$. (c) On a single set of axes, sketch $y = f(x)$, $y = f^{-1}(x)$ and $y = x$. State the symmetry.
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12**Mini-investigation — natural domain.** Three students propose different formulas for the same function: - Anya: $f(x) = \dfrac{x^2 - 1}{x - 1}$ - Bao: $f(x) = x + 1$ - Cara: $f(x) = x + 1$ for $x \neq 1$ (a) Compute $f(2)$ 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.
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