11.3 Functions

Let $f(x) = 3x + -1$. Find $f(4)$.

For $f(x) = 3x + -2$, find $x$ such that $f(x) = 10$.

State the largest natural domain over $\mathbb{R}$ of $f(x) = \dfrac{1}{x - -2}$.

State the largest natural domain over $\mathbb{R}$ of $g(x) = \sqrt{x - 5}$.

Let $f(x) = x^2 + 2x + -5$. Find $f(3)$.

A vertical line drawn through the graph of $y = x^2$ meets the curve in at most one point. Is $y = x^2$ a function of $x$?

State the range of $f(x) = -3x + 4$ defined on $\mathbb{R}$.

For $f(x) = 3x - 2$, find the image of $x = -2$.

State the largest natural domain of $h(x) = \dfrac{1}{\sqrt{x - 4}}$.

The function $f$ is defined by the table: $f(1) = 4$, $f(2) = 7$, $f(3) = 10$. Find a formula for $f(n)$ if it is linear.

Let $f(x) = 2x + 5$ and $g(x) = x^2 - 3$. Find $f(g( 4))$.

Let $f(x) = 2x + 5$ and $g(x) = x^2 - 3$. Find a simplified expression for $(f \circ g)(x)$.

Find $f^{-1}(x)$ for $f(x) = 3x + -1$.

State the domain and range of $f(x) = \dfrac{1}{x - 2}$.

For $f(x) = 2x + 5$ and $g(x) = -x + 8$, solve $f(x) = g(x)$.

State the domain and range of $f(x) = |x - 3|$.

For what values of $x$ is $f(x) = \sqrt{2x - 6}$ defined?

For $f(x) = x^2$ and $g(x) = x + 6$, find all $x$ with $f(x) = g(x)$.

If $f(3) = 7$ and $f$ is one-to-one, state $f^{-1}(7)$.

Let $f(x) = \begin{cases} 2x + 1 & x < 0 \\ x^2 & x \geq 0 \end{cases}$. Find $f(-3)$ and $f(2)$.

Let $f(x) = 4x + -3$. Find $f^{-1}(x)$ and state its domain.

Let $f(x) = \sqrt{x}$ and $g(x) = x - 4$. State $(f \circ g)(x)$ and its largest natural domain.

State the range of $f(x) = |x - 3| + 2$.

For $f(x) = \lfloor x \rfloor$ (the floor function), find $f(2.7)$ and $f(-1.4)$.

$f(x) = x^2$ is not invertible on $\mathbb{R}$. State a domain restriction that makes it invertible, and give the inverse.

A graph passes through $(0, 4)$ and decreases monotonically, approaching $y = 0$ but never reaching it. Is this consistent with $f(x) = 4 \cdot (0.5)^x$? Justify.

Let $f(x) = 2x + 1$ and $g(x) = \dfrac{1}{x}$ for $x \neq 0$. Find $(g \circ f)(x)$ and state its domain.

For $f(x) = |2x - 4|$, solve $f(x) = 6$.

The cost (CHF) of producing $n$ bottles is $C(n) = 0.5n + 200$. State (a) the meaning of the gradient, (b) the meaning of $C(0)$.

The graph of $y = f^{-1}(x)$ is the reflection of the graph of $y = f(x)$ in which line?

Let $f(x) = 2x + 1$ and $g(x) = \dfrac{x}{x - 1}$ for $x \neq 1$. Find $(g \circ f)(x)$ and state its domain.

For $f(x) = \dfrac{1}{x - 1}$, find $(f \circ f)(x)$ and identify the values that are fixed by $f \circ f$.

Find the inverse of $f(x) = \dfrac{2x + 3}{x - 1}$ for $x \neq 1$, and state its domain.

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

For $f(x) = \begin{cases} 2x + a & x < 1 \\ x^2 & x \geq 1 \end{cases}$, find $a$ such that $f$ is continuous at $x = 1$.

Find the range of $f(x) = x^2 - 4x + 7$ on $\mathbb{R}$.

The graph of $y = f(x)$ passes through $(1, 4)$ and $(3, 10)$. If $f$ is linear, find $f^{-1}(x)$.

Let $f(x) = 2x - 1$. Solve $f(f(x)) = 11$.

Let $f(x) = (x - 2)^2$ for $x \geq 2$. Find $f^{-1}(x)$ and its domain.

A circle of radius 5 centred at the origin has equation $x^2 + y^2 = 25$. Explain why this is **not** a function of $x$, and write the two functions $y = f_1(x)$ and $y = f_2(x)$ that together describe the circle.