Mathematics
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Fluency · Pack A

Functions

Answer each question. Show working where needed.

Bronze

  1. Given $f(x) = 3x + 2$, find $f(4)$.

  2. Given $f(x) = x^2 + 1$, find $f(3)$.

  3. Given $f(x) = 2x + 5$, find $f(-3)$.

  4. Given $f(x) = 3x + 2$, find $x$ such that $f(x) = 17$.

  5. A function maps $x$ to $2x + 3$. Find the image of $5$.

  6. A function maps $x$ to $3x - 1$. What value of $x$ maps to $14$?

  7. A function $f$ has domain $\{1, 2, 3, 4, 5\}$ and rule $f(x) = 3x$. List the range.

  8. Given $g(x) = x^2 - 4$, find $g(0)$ and $g(3)$.

  9. A linear function maps $0 \to 5$ and $1 \to 8$. Find $f(x)$.

  10. A function machine multiplies by $4$ then adds $7$. Write the function $f(x)$.

Silver

  1. State the domain and range of $f(x) = 2x + 3$ where $x \in \mathbb{R}$.

  2. Find the range of $f(x) = x^2 + 3$ for $x \in \mathbb{R}$.

  3. A function $f(x) = 3x + 1$ has domain $0 \leq x \leq 4$. State the range.

  4. Given $f(x) = x + 3$ and $g(x) = 2x$, find $(f \circ g)(4)$.

  5. Given $f(x) = 2x + 1$ and $g(x) = x + 3$, find $(f \circ g)(x)$.

  6. Given $f(x) = x^2$ and $g(x) = x + 1$, find (a) $(f \circ g)(3)$ and (b) $(g \circ f)(3)$.

  7. Find the inverse of $f(x) = 3x + 2$.

  8. Given $f(x) = 2x + 1$, find $f^{-1}(9)$.

  9. A function $f$ is defined by $f: x \mapsto 2x - 6$. Find $f(5)$ and an $x$ with $f(x) = 0$.

  10. A linear function $f$ satisfies $f(1) = 5$ and $f(4) = 14$. Find $f(x)$.

Gold

  1. Given $f(x) = 3x + 1$ and $g(x) = x^2$, find $(f \circ g)(x)$.

  2. Given $f(x) = 2x + 3$ and $g(x) = x^2$, find $(g \circ f)(x)$.

  3. State the largest possible domain of $f(x) = \sqrt{x - 4}$.

  4. State the largest possible domain of $f(x) = \dfrac{1}{x - 3}$.

  5. Given $f(x) = x^2$ and $g(x) = 3x + 4$, solve $f(x) = g(x)$.

  6. Find the inverse of $f(x) = \dfrac{x + 3}{2}$.

  7. Show that for $f(x) = 3x - 1$ and $f^{-1}(x) = \dfrac{x + 1}{3}$, we have $(f \circ f^{-1})(7) = 7$.

  8. Given $f(x) = x + 3$, $g(x) = 2x$, $h(x) = x^2$, find $(f \circ g \circ h)(2)$.

  9. Given $f(x) = \sqrt{x}$ and $g(x) = x + 5$, find $(f \circ g)(11)$.

  10. Find the inverse of $f(x) = (x - 3)^2$ for $x \geq 3$.

Platinum

  1. Given $f(x) = 2x + 1$ and $g(x) = x + 3$, find $(f \circ g)^{-1}(x)$.

  2. Find the largest possible domain of $f(x) = \dfrac{1}{\sqrt{x - 4}}$.

  3. A car rental costs a £$25$ fixed fee plus £$12$ per day. (a) Write $C(n)$ for $n$ days. (b) Find $C^{-1}(c)$ and interpret.

  4. A function $f$ has graph $y = (x - 3)^2 + 2$. Find the (a) minimum value and (b) range of $f$.

  5. For $f(x) = 2x - 1$ and $g(x) = x^2 + 3$, solve $(f \circ g)(x) = 13$.

  6. A function $f(x) = ax + b$ satisfies $f(0) = 3$ and $f^{-1}(11) = 2$. Find $a$ and $b$.

  7. For $f(x) = x + 1$, find $f(f(f(x)))$.

  8. Given $f(x) = 3x + 2$, find the value of $x$ for which $f(x) = f^{-1}(x)$.

  9. A water tank has volume function $V(t) = 8t + 20$ litres after $t$ minutes ($t \geq 0$). After how many minutes does the tank contain $100$ litres?

  10. For $f(x) = \sqrt{x}$ (domain $x \geq 0$) and $g(x) = x - 4$, find the domain of $(f \circ g)(x)$.