Functions

The lines $L_1: y = mx + c$ and $L_2: y = nx + d$ are perpendicular. If $m = 3$, find the value of $n$.

Find the value of $k$ such that the point $(2, k)$ lies on the line $y = 4x - 1$.

Line $L_2$ is parallel to $y = 5x - 7$. What is the gradient of $L_2$?

Determine the length of the segment connecting $(1, 1)$ and $(4, 5)$.

A line $L_1$ passes through the points $A(-2, 5)$ and $B(4, 2)$. A second line $L_2$ is perpendicular to $L_1$ and passes through the point $C(1, 1)$. Find the equation of line $L_2$, expressing it in the form $y = mx + c$.

In the diagram, $ABCD$ is a rhombus. If $A$ is $(1, 2)$ and $C$ is $(5, 6)$, what is the gradient of the diagonal $BD$?

Line $L$ has an $x$-intercept of $5$ and a $y$-intercept of $-2$. Find the equation of $L$ in the form $ax + by + d = 0$ where $a, b, d$ are integers.

Two drones, $A$ and $B$, travel along straight paths. Drone $A$ follows the path $y = 3x + 1$. Drone $B$ follows a path that is parallel to Drone $A$ and passes through the point $(2, 10)$. Find the distance between the two paths (the perpendicular distance between the lines).

A line $L$ is tangent to a circle at the point $(3, 4)$. The circle is centered at the origin $(0, 0)$. Find the $x$-intercept of line $L$.

The population of a small town increases linearly. In the year $2010$ ($t=0$), the population was $12,000$. In $2018$ ($t=8$), the population was $14,560$. If this trend continues, in what year will the population reach $17,120$?

The diagram shows a parabola with roots at $x = -2$ and $x = 4$. What is the $x$-coordinate of the vertex?

What is the range of the function $f(x) = (x-1)^2 + 4$?

If the graph of $f(x) = x^2$ is translated $3$ units to the right and $2$ units up, what is the new equation?

A rectangle has a perimeter of $20$ m. If one side is $x$, the area is $A(x) = x(10-x)$. What value of $x$ maximizes the area?

A rectangular garden is fenced on three sides using $40$ m of wire, with the fourth side being a wall. If the width is $x$, find the maximum area.

If $f(x) = x^2 + px + q$ has a vertex at $(-2, 5)$, what is the value of $q$?

The quadratic function $f(x) = kx^2 + 4x + k$ has two equal real roots. Given $k > 0$, find $k$.

Consider the function $f(x) = ax^2 + bx + c$. The sum of the roots is $6$ and the product of the roots is $4$. If the graph passes through $(1, 9)$, find the value of $a$.

A path is built around a rectangular garden of $8$ m by $12$ m. The path has a uniform width $w$. If the total area of the garden and the path combined is $165$ $m^2$, find the width $w$ in meters.

The function $g(x) = kx^2 + (k-3)x + 1$ is always positive for all $x \in \mathbb{R}$. Determine the range of possible values for $k$.

The graph shows $f(x) = \log_{10}(x)$. What is the domain of this function?

Identify the $x$-intercepts of the function $f(x) = (x-1)(x+3)$ from the provided graph.

For the function $f$ illustrated in the mapping diagram, which element of the domain maps to $9$?

The graph shows a transformation of $y = x^2$. Identify the coordinates of the vertex.

The graph shows a cubic function $f(x) = x^3 - 3x$. How many real roots does the function have?

The function $f(x) = 2^x - 4$ is graphed. Determine the coordinates of the $x$-intercept.

The graph of $f(x) = |x-2| - 1$ is shown. Find the $x$-intercepts.

A mapping $f$ transforms a coordinate $x$ to $f(x) = \frac{2x+1}{x-3}$. Find the value of $x$ that must be excluded from the domain and the value of $y$ that is excluded from the range.

The graph of $y = f(x)$ is a semicircle with endpoints $(-4, 0)$ and $(4, 0)$. The function $g(x)$ is defined as $g(x) = f(2x) + 3$. What is the range of $g(x)$?

A production cost function is given by $C(n) = \frac{500n + 1000}{n+5}$, where $n$ is the number of units produced. As production $n$ increases indefinitely, what value does the average cost per unit approach?

The functions $f$ and $g$ are defined for $x \in \mathbb{R}$ by $f(x) = 2x - 3$ and $g(x) = x^2 + 1$. The diagram below illustrates the relationship between the inputs and outputs of the composite function $h(x) = (g \circ f)(x)$. Find the value of $(g \circ f)(4)$.

If $f(x) = \ln(x)$ and $g(x) = e^{2x}$, find $(f \circ g)(x)$.

Let $f(x) = 4 - x$. Find $(f \circ f \circ f)(x)$.

Find $(g \circ f)(x)$ for $f(x) = x + 2$ and $g(x) = 3x^2$.

The graph of a linear function $f(x) = ax + b$ is shown in the coordinate plane below. Given that $g(x) = 3x - 1$ and $(f \circ g)(x) = 6x - 1$, determine the value of the constant $b$.

Let $f(x) = \cos(x)$. Find $(f \circ f)(0)$ correct to 3 decimal places (radians).

If $f(x) = x+3$ and $g(x) = x^2$, solve $(g \circ f)(x) = 16$.

Two functions are defined as $f(x) = \frac{1}{x}$ and $g(x) = \cos(x)$. For what value of $x$ in the interval $0 < x < \pi$ does $(f \circ g)(x) = 2$?

The velocity $v$ of a particle in m/s is given by $v(s) = \sqrt{2s + 9}$, where $s$ is the displacement in meters. The displacement $s$ varies with time $t$ as $s(t) = t^2 + 4t$. Determine the velocity of the particle at $t = 2$ seconds by evaluating $(v \circ s)(2)$.

Let $f(x) = 2^x$ and $g(x) = \log_2(x^2 - 1)$. Solve for $x$ in the equation $(f \circ g)(x) = 8$.

The following mapping diagram represents a one-to-one function $h$ that maps elements from set $A$ to set $B$. If $h(3) = 7$, find the value of $h^{-1}(7)$.

The function $g(x) = \sqrt{x-1}$ is defined for $x \geq 1$. The graph of $g$ is shown below. Determine the domain of the inverse function $g^{-1}$.

The diagram shows the graph of a linear function $f(x) = 2x - 4$. The inverse function, $f^{-1}(x)$, is also a linear function. Find the expression for $f^{-1}(x)$.

A function $f$ is defined as $f(x) = 5 - x$. What is $f^{-1}(x)$?

If $f(x) = \frac{x}{x+1}$, show that $f^{-1}(x) = \frac{x}{1-x}$. What is the value of $f^{-1}(\frac{1}{2})$?

Given $f(x) = 2 + \sqrt{x-1}$, find the range of $f^{-1}$.

The function $f(x) = \log_{2}(x+1)$. Find the value of $f^{-1}(3)$.

A mapping for a specific geometric transformation is given by $f(x) = \ln(x^2 - 1)$ for $x > 1$. To reverse this transformation, we find the inverse function. Which of the following defines $f^{-1}(x)$ and its domain?

Given $f(x) = x^3 + 2x + 1$, it is known that $f$ is a strictly increasing function and therefore has an inverse $f^{-1}$. If $g(x) = f^{-1}(x)$, find the gradient of the tangent to the curve $y = g(x)$ at the point where $x = 13$.

An object's distance $d$ (in meters) from a sensor is given by $d(t) = \frac{10}{t+1}$ for $t \ge 0$. A computer needs to calculate the time $t$ for any given distance $d$ in the sensor's range. Determine the rate of change of the inverse function $t'(d)$ when the object is 2 meters away.

The function $f(x) = x^2$ is transformed to $g(x) = 2x^2$. Describe the transformation.

The graph of $y = f(x)$ is shown. Which of the following is the graph of $y = -f(x)$?

If $f(x) = x^3$, what is the equation of the graph after a horizontal stretch with scale factor $4$?

Which transformation maps the graph of $y = f(x)$ onto $y = f(x - 4)$?

The graph of the function $y = f(x)$ is shown in the diagram below. The function is transformed to create a new function $g(x) = f(x - 2) + 1$. The point $P(0, 2)$ lies on the graph of $f$. What are the coordinates of the corresponding point $P'$ on the graph of $g$?

A function $f(x)$ is symmetric about the $y$-axis. Which of the following transformations would preserve this symmetry?

If $f(x) = 2^x$, what transformation results in $g(x) = 2^{x-3} + 1$?

The temperature $T$ in a kiln is $T(h) = 20 + 150\sqrt{h}$ where $h$ is hours. To calibrate for a new sensor that records time in minutes $m$, and reads 10 degrees higher than the actual temperature, what is the sensor's function $S(m)$?

A geometric pattern follows the curve $y = \sin(x)$. To fit a specific design, the pattern is compressed horizontally by a factor of 3, stretched vertically by a factor of 2, and shifted $\pi/6$ units to the right. What is the new equation?

A population of bacteria grows according to $P(t) = 100 e^{0.1t}$. A drug is introduced that reduces the growth rate by half and immediately kills 20% of the existing bacteria. Which function describes the new population $Q(t)$?

If $f(x) = 2^x + 3$, what is the horizontal asymptote?

What is the vertical asymptote of $f(x) = 3 + \frac{1}{x}$?

Determine the horizontal asymptote of $f(x) = \frac{1-3x}{x+2}$.

The function $f(x) = \frac{2x-1}{x+k}$ has a vertical asymptote at $x = -4$. Find the value of $k$.

Consider the function $f(x) = \frac{ax + 5}{2x - b}$, where $a, b \in \mathbb{R}$. The graph of $y = f(x)$ is shown below, featuring a vertical asymptote at $x = 3$ and a horizontal asymptote at $y = 2$. Determine the values of $a$ and $b$.

Identify the vertical asymptote of $f(x) = \tan(x)$ in the interval $0 < x < \pi$.

For the function $f(x) = \frac{2x+3}{x-1}$, what is the point of intersection of the vertical and horizontal asymptotes?

Find the equation of the vertical asymptote for the function $f(x) = \ln(2x - 8) + 5$.

The total profit $P$ (in thousands) of a startup is given by $P(x) = \frac{20x^2 - 100}{x^2 + 4}$, where $x$ is the number of months since launch. What is the value of the horizontal asymptote, and after how many months (to the nearest month) does the profit reach 90% of this asymptotic value?

A mountain climber's oxygen level $L$ (percentage of sea level) at altitude $h$ (in km) is modeled by $L(h) = \frac{100}{1 + 0.15h}$. Although the climber cannot reach 'infinite' altitude, what is the horizontal asymptote of this model and its physical significance?

The graph of the exponential function $f(x) = a^x$ passes through the point $P(2, 25)$, where $a > 0$. Find the value of $f(3)$.

In the function $y = A \cdot b^x$, if the graph passes through $(0, 7)$ and $(1, 21)$, find the value of $b$.

What is the range of the function $f(x) = 2^x + 5$?