Functions: Mathematics 9709 (Cambridge International AS & A Level)

The function $h$ is defined, for $x \le 4$, by $h(x) = (x - 4)^2 + 1$.

What is the range of $h$?

The functions $f$ and $g$ are defined, for $x \in \mathbb{R}$, by $f(x) = 2x - 1 \qquad \text{and} \qquad g(x) = x^2 + 2.$

(a) Find $fg(x)$, simplifying your answer. [2]

(b) Find $gf(x)$, simplifying your answer. [2]

(c) Hence, or otherwise, find the value(s) of $x$ for which $fg(x) = gf(x)$. [3]

The function $f$ is defined, for $x \in \mathbb{R}$, $x \ne 1$, by $f(x) = \frac{3x+2}{x-1}.$

(a) Find $f^{-1}(x)$, showing your working clearly. [3]

(b) State the domain of $f^{-1}$. [1]

(c) By first finding $f(2)$, verify that $f^{-1}f(2) = 2$. [2]

The graph of $y = f(x)$ has a single minimum point at $(4, -3)$.

(a) State the coordinates of the minimum point on the graph of $y = f(x) + 6$. [1]

(b) State the coordinates of the minimum point on the graph of $y = f(x-2)$. [1]

(c) State the coordinates of the minimum point on the graph of $y = 2f(x) - 1$. [2]

(d) State the coordinates of the minimum point on the graph of $y = f(3x)$. [2]

The function $f$ is defined, for $x \in \mathbb{R}$, by $f(x) = (x+1)^2 - 4$.

On its natural domain $x \in \mathbb{R}$, $f$ is not one-one, for example $f(-3) = f(1) = 0$, so it has no inverse there.

What is the largest possible domain, containing $x=0$, on which $f$ is one-one?

The function $f$ is defined by $f(x) = \sqrt{3x - 12}$.

What is the domain of $f$?

The functions $f$ and $g$ are defined by $f(x) = \frac{1}{x-3}, \quad x \in \mathbb{R}, \ x \ne 3 \qquad \text{and} \qquad g(x) = 2x+1, \quad x \in \mathbb{R}.$

(a) Find $fg(x)$, simplifying your answer as a single fraction. [2]

(b) State the domain of $fg$. [2]

(c) Find $gf(x)$, simplifying your answer. [2]

(d) State the domain of $gf$. [1]

The function $f$ is defined by $f(x) = x^2 - 6x + 11, \quad x \in \mathbb{R}, \ x \ge 3.$

(a) Express $f(x)$ in the form $(x-a)^2 + b$, and hence state the range of $f$. [3]

(b) Find $f^{-1}(x)$, showing your working clearly. [3]

(c) State the domain of $f^{-1}$. [1]

(d) Solve the equation $f^{-1}(x) = 7$. [2]

The functions $f$ and $g$ are defined, for $x \in \mathbb{R}$, by $f(x) = x^2 - 1 \qquad \text{and} \qquad g(x) = 3 - x.$

What is the value of $gf(-2)$?

The graph of $y = f(x)$ passes through the point $(6, -2)$.

(a) State the coordinates of the corresponding point on the graph of $y = f(x) - 5$. [1]

(b) State the coordinates of the corresponding point on the graph of $y = f(x+3)$. [1]

(c) State the coordinates of the corresponding point on the graph of $y = \frac{1}{2}f(x)$. [1]

(d) State the coordinates of the corresponding point on the graph of $y = 4f(x+3) - 1$. [2]