Functions

The relationship between the number of hours a cooling fan is left running, $x$, and the total energy consumed in kilowatt-hours, $E$, is represented by a linear function $E(x) = 0.15x + 0.5$. The initial $0.5$ kWh represents the energy used during the startup phase. Based on the provided graph, determine the energy consumed if the fan runs for exactly $8$ hours.

A student models the height of a bouncing ball using the function $h(t) = -4.9t^2 + 10t + 2$, where $h$ is the height in meters and $t$ is the time in seconds since the ball was thrown. The graph of this function is shown below. Which of the following best describes the domain of this function in the context of the physical situation?

Identify the range of the function $f(x)$ depicted in the provided graph.

An vertical line test is applied to the curve shown. What does the test conclude?

The graph of $y = f(x)$ is transformed to $y = f(x) + k$. If the new graph passes through $(0, 5)$, what is the value of $k$?

Consider the composite function flow. If $f(x) = x + 2$ and $g(x) = 3x$, what is the final output $Z$ when $x = 4$?

The graph shows $f(x) = a^x + b$. Determine the value of $b$ based on the horizontal asymptote.

A cable hangs between two poles $20$ meters apart. The height $h$ of the cable above the ground at a distance $x$ from the left pole is modeled by $h(x) = 0.1x^2 - 2x + 15$ for $0 \leq x \leq 20$. Find the minimum height of the cable and the distance $x$ where this occurs.

A sound engineer uses the function $L(I) = 10 \log(\frac{I}{10^{-12}})$ to measure sound level $L$ in decibels (dB) for an intensity $I$. If the intensity is doubled, what is the approximate increase in the sound level?

A projectile is launched from a platform $10$ meters high with an initial velocity. Its height $H$ in meters after $t$ seconds is $H(t) = -4.9t^2 + 20t + 10$. A wall $15$ meters high is located $30$ meters away from the launch site. If the horizontal distance $x$ is given by $x(t) = 15t$, determine if the projectile clears the wall and what its height is when it reaches the horizontal distance of $30$ meters.

A piece of machinery depreciates in value according to $V(t) = 5000(0.8)^t$, where $t$ is in years. If the machinery is used for 10 years, what is the range of its value (to the nearest dollar)?

The graph shows the depth of water in a tidal harbor over 12 hours. Which interval represents the range of the depth $D$?

A rectangle has a fixed perimeter of 20 cm. If $x$ is the width, the area is $A(x) = x(10 - x)$. Given that $x$ must be a positive length and the area must be positive, what is the domain of $A(x)$?

Consider the function $f(x) = 4 - x^2$ restricted to the domain $-1 \le x \le 3$. Based on the graph, what is the range of the function?

The profit $P$ (in thousands of dollars) for a company is modeled by $P(x) = -2x^2 + 12x - 10$, where $x$ is the number of units sold (in hundreds). What is the range of the profit function for the units $x$ where the company makes a profit ($P \ge 0$)?

A drone's altitude $y$ in meters relative to its horizontal distance $x$ from a controller is $y = \sqrt{10000 - x^2}$. What is the physical domain of $x$ for which the drone is above or at ground level?

A piece of wire 40 cm long is bent into a sector of a circle with radius $r$ and angle $\theta$ (in radians). The perimeter is $P = 2r + r\theta = 40$. The area is $A = 0.5r^2\theta = 0.5r^2(\frac{40-2r}{r}) = 20r - r^2$. If the angle $\theta$ must be between 0 and $\pi$, what is the domain of the radius $r$?

A cable hangs between two poles such that its height $y$ (metres) at a horizontal distance $x$ (metres) from the left pole is $y = 0.1x^2 - 2x + 15$ for $0 \le x \le 20$. Find the range of heights of the cable.

A piece of wire 40 cm long is cut into two pieces. One piece is bent into a square of side $s$ and the other into a circle. The total area $A(s)$ of the two shapes is a function of $s$. If the square must have a side length of at least 2 cm and at most 8 cm, find the range of the total area $A(s)$, to the nearest whole number.

An athlete's heart rate $R$ (bpm) during a 20-minute workout is modeled by $R(t) = -0.4t^2 + 8t + 70$, where $t$ is the time in minutes ($0 \le t \le 20$). What is the range of the athlete's heart rate during this period?

Identify which of the following diagrams represents a function where each input $x$ has exactly one output $y$.

For the function $f(x) = 2x + k$, the graph passes through the point $(1, 7)$. Find the value of $k$.

Given $h(x) = \frac{12}{x}$, find the value of $x$ for which $h(x) = 3$.

A function is defined as $y = f(x)$. If the point $(3, 10)$ lies on the graph of the function, which of the following statements must be true?

A small business produces eco-friendly water bottles. The total daily cost, $C$, in euros (EUR), of producing $x$ bottles is modeled by the function $C(x) = 0.5x^2 - 10x + 300$, where $10 \le x \le 50$. The graph of this function is shown below. Using this model, calculate the total daily cost if the business produces 40 bottles.

If $f(x) = 3x - 1$, find the inverse function $f^{-1}(x)$.

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

The height of a drone, $H$ meters above the ground, is modeled by the function $H(x) = 15 + 10\sin(0.2x)$, where $x$ is the horizontal distance in meters from a start point. Due to a change in the wind, the drone's flight path is adjusted. The new height function is defined as $G(x) = H(x - 5) + 2$. Using the provided graph of $H(x)$, determine the value of $G(10\pi + 5)$.

A chemical reaction produces heat, and the temperature, $T$, in degrees Celsius ($^{\circ}\text{C}$) of the solution is modeled by the function $T(t) = -0.5t^2 + 6t + 20$, where $t$ is the time in minutes after the catalyst is added. The graph of $T(t)$ is shown below for the interval $0 \le t \le 12$. A scientist needs to find the exact time when the temperature reaches its maximum value. Using function notation, let $t_{max}$ be the time such that $T(t_{max})$ is the maximum temperature. Calculate the value of $T(2t_{max} - 10)$.

A sound's intensity level $L$ in decibels is given by $L(I) = 10 \log_{10}(\frac{I}{10^{-12}})$, where $I$ is the intensity in $W/m^2$. If a specific sound has an intensity of $I = 2.5 \times 10^{-4} W/m^2$, find the intensity level $L$ rounded to the nearest decibel.

A taxi service charges a fixed base fare plus a constant rate per kilometer traveled. The total cost, $C$, in dollars, for a journey of $d$ kilometers is shown in the graph below.

Find the gradient (slope) of the line and state what it represents in the context of the problem.

If $f(x) = mx + c$, $f(0) = 4$ and $f(3) = 13$, find the value of $m$.

What is the gradient (slope) of the line $3x + 4y = 12$?

A printer has $2000$ sheets of paper. It prints at a constant rate of $40$ pages per minute. How long will it take until only $200$ sheets remain?

Find the value of $k$ such that the line $y = kx + 4$ is parallel to the line $2x - 5y = 10$.

A linear model for the population $P$ of a town $t$ years after $2010$ is $P = 500t + 12000$. In what year will the population reach $20000$?

The line $L$ has equation $y = 0.5x + c$ and passes through the midpoint of the segment joining $(2, 4)$ and $(6, 12)$. Find $c$.

A salesperson earns a base salary plus a commission on every dollar of sales. If they sell $10,000 worth of goods, their total pay is$2,800. If they sell $25,000, their total pay is$4,300. Calculate their total pay if they sell $40,000.

A spring has an original length of 12 cm. When a mass of 5 kg is attached, the length becomes 15.5 cm. Assuming the extension follows a linear model (Hooke's Law), what mass is required to stretch the spring to a total length of 20.4 cm?

A candle is 30 cm long when lit. After 40 minutes of burning, its length is 22 cm. Assuming the candle burns at a constant rate, what will its length be after 100 minutes of burning?

Find the equation of the axis of symmetry for the parabola $y = 2x^2 - 8x + 3$.

The path of a diver is modeled by $h(x) = -x^2 + 2x + 8$, where $h$ is the height above the water and $x$ is the horizontal distance from the diving board. At what horizontal distance $x$ does the diver hit the water ($h=0$)?

A function is defined by $f(x) = (x+3)^2 - 4$. What are the coordinates of the vertex of the graph of $f$?

The graph of $y = x^2 + bx + 9$ touches the $x$-axis at exactly one point. Given that $b > 0$, find the value of $b$.

The production cost of a factory is $C(x) = x^2 - 10x + 30$, where $x$ is the number of machines running. What is the minimum cost?

A ball is kicked from a point $O$ on the ground. The path is $y = x - 0.05x^2$ where $y$ is the height and $x$ is the horizontal distance. What is the horizontal distance between the kicking point and the point where the ball hits the ground?

Given the quadratic function $f(x) = x^2 + kx + 4$. For what values of $k$ does the graph of $f$ not intersect the $x$-axis?

A quadratic model $y = ax^2 + bx + c$ is used to fit the data points representing the shape of a cooling tower's side: $(0, 10)$, $(10, 6)$, and $(20, 10)$. Using these points, determine the value of the coefficient $a$.

An archer fires an arrow such that its height $h$ in metres is given by $h(d) = -0.01d^2 + 0.4d + 1.5$, where $d$ is the horizontal distance in metres. A target is placed at a distance $d$ where the arrow is at its maximum height. How high is the center of the target?

A company’s weekly production cost $C$ in euros is $C(x) = 0.5x^2 - 40x + 2000$, where $x$ is the number of units produced. To minimize the average cost per unit, $AC(x) = \frac{C(x)}{x}$, the company looks for the vertex of the cost function. However, they first want to find the production level $x$ that minimizes the total cost $C(x)$. Find this value of $x$.

Identify the horizontal asymptote of the function $f(x) = 5(0.2)^x - 3$.

An exponential function is given by $y = 10 \cdot e^{0.5x}$. Calculate the value of $y$ when $x = 2$.

The intensity of light $I$ passing through a filter is given by $I = I_0 (0.9)^d$, where $d$ is the thickness in mm. If the original intensity $I_0$ is 200 units, find the intensity after passing through 5 mm.

A culture of 100 cells increases by $15\%$ every hour. Which function represents the number of cells $C$ after $t$ hours?

A biologist is monitoring the population of a specific strain of bacteria in a laboratory petri dish. The population $P$ of the bacteria $t$ hours after the experiment begins is modeled by the function $P(t) = a \cdot b^t$, where $a$ and $b$ are positive constants. The graph of the population over the first 5 hours is shown below. Given that the initial population was 150 bacteria and after 2 hours the population increased to 600 bacteria, find the population of the bacteria after 5 hours.

Given the function $h(x) = 12 - 4e^{0.2x}$, find the value of $x$ for which $h(x) = 0$.

A bank offers a savings account with $3.5\%$ interest per annum compounded monthly. If $P$ is the principal, the amount $A$ after $t$ years is $A = P(1 + \frac{0.035}{12})^{12t}$. How many years will it take for the investment to increase by $50\%$?

A cup of coffee at $90^{\circ}$C is placed in a room with a constant temperature of $20^{\circ}$C. The temperature of the coffee $T$ after $t$ minutes is $T(t) = 20 + 70(0.92)^t$. How many minutes does it take for the coffee to reach $45^{\circ}$C?

The atmospheric pressure $P$ (in kPa) at an altitude $h$ (in km) is given by $P(h) = 101.3 e^{-0.125h}$. At what altitude is the pressure exactly half of the pressure at sea level ($h=0$)?

A certain population of rodents grows such that the growth rate is proportional to the current population. If the population doubles every $8$ months and starts at $500$, what will the population be after $3$ years?

1. Logarithmic functions (HL): practice question 1 for Functions?

1. Logarithmic functions (HL): practice question 2 for Functions?

1. Logarithmic functions (HL): practice question 3 for Functions?

1. Logarithmic functions (HL): practice question 4 for Functions?

1. Logarithmic functions (HL): practice question 5 for Functions?

1. Logarithmic functions (HL): practice question 6 for Functions?

1. Logarithmic functions (HL): practice question 7 for Functions?

1. Logarithmic functions (HL): practice question 8 for Functions?

1. Logarithmic functions (HL): practice question 9 for Functions?

1. Logarithmic functions (HL): practice question 10 for Functions?

A wave is described by $y = \sin(B x)$. If the period of the wave is $10\pi$, what is the value of $B$?

In the function $y = 5 \sin(x) - 3$, what is the minimum value of $y$?

The blades of a wind turbine rotate at a constant speed. The height of the tip of one blade is modeled by $h(t) = a \cos(bt) + 60$. If the maximum height is $100$ m, what is the value of $a$?

For the function $f(x) = 10 \sin(x) + d$, the range is $[2, 22]$. Find the value of $d$.

The height, $H$ metres, of the tide in a coastal harbour can be modelled by the function $H(t) = a \cos(b t) + d$, where $t$ is the number of hours after midnight. The graph shows the height of the tide over a 24-hour period. High tide occurs at midnight with a height of $11$ m, and low tide occurs $6$ hours later with a height of $3$ m. Find the value of the parameter $b$ in radians per hour.

A Ferris wheel starts at its midline at $t=0$ and is moving upwards. The wheel has a radius of 10m and a midline at 12m. It takes 8 minutes for one revolution. Which function represents this?

The height of water in a tank is $h(t) = 3 \sin(0.1t) + 5$. Find the rate of change of the height when $t = 10$ using your GDC (derivative).

A mechanical piston moves such that its distance $x$ in cm from a reference point is $x(t) = 15 \cos(10t) + 20$, where $t$ is in seconds. Find the total distance traveled by the piston in the first $\frac{\pi}{5}$ seconds.

The depth of water in a tank fluctuates sinusoidally over time due to an automated system. The depth $y$ (cm) at time $x$ (minutes) is shown in the graph below. Based on the graph, which equation correctly models the depth of the water?

A musician plays a pure tone that creates a pressure wave modeled by $P(t) = 0.02 \sin(880\pi t)$, where $P$ is pressure in Pascals and $t$ is time in seconds. What is the frequency of this sound wave in Hertz (Hz)?

A storage facility charges $100$ per month for the first $50$ $m^2$ of space and then $1.50$ for each additional $m^2$. A customer rents $85$ $m^2$. What is the total monthly charge?

The piecewise function $f(x)$ is defined as: $f(x) = \begin{cases} 2 & x \le 1 \\ x + 1 & x > 1 \end{cases}$ Find $f(1)$.

A phone plan has a monthly cost of $20$ for up to $500$ MB of data. After $500$ MB, the cost increases by $0.05$ for every additional MB. Which of the following represents the total cost $C$ for $x$ MB of data where $x > 500$?

A piecewise function $h(x)$ is defined below. Based on the diagram, what is the value of $h(0)$? $h(x) = \begin{cases} 3x + 1 & x < 0 \\ -x + 4 & x \ge 0 \end{cases}$

A logistics company calculates the shipping cost, $C$ (in dollars), for a package based on its weight, $w$ (in kg). The cost follows a piecewise linear model. For weights up to and including $2$ kg, the cost is a flat fee of $10$. For weights greater than $2$ kg, the cost increases at a rate of $2.50$ per kg for every kg above $2$. The graph of the cost function is shown below.

Which of the following expressions correctly defines the cost function $C(w)$ for $w > 0$?

A piecewise function is defined as: $f(x) = \begin{cases} 2^x & x < 2 \\ kx - 2 & x \ge 2 \end{cases}$ If the function is continuous at $x = 2$, what is the value of $k$?