Representing Functions

Find the domain of the radical function $f(x) = \sqrt{2x - 8}$.

The period of oscillation $T$ (in seconds) of a simple pendulum depends on its length $L$ (in feet) according to the radical function $T(L) = 2\pi\sqrt{\frac{L}{9,8}}$. Calculate the domain of the function in this context and determine the period of a pendulum of $2,45$ feet in length.

A taxi driver charges an initial fee of $\$3$ and $\$1,5$ for each mile traveled. Write the algebraic expression of the function that relates the cost of the trip to the miles traveled.

Observe the graph of the line and determine its slope, its y-intercept, and its algebraic expression.

Determine the coordinates of the vertex and the equation of the axis of symmetry of the parabola $f(x) = x^2 - 6x + 5$.

From the parent parabola $y = x^2$ (dashed line), a translation was applied to obtain the parabola drawn with a solid line. Write the equation of the new parabola and state the translation vector applied.

The profit of a technology company, in thousands of dollars, is given by the function $B(x) = -2x^2 + 120x - 1000$, where $x$ is the number of devices sold in hundreds. How many devices must they sell to maximize their profit and what is this maximum profit?

A farmer has $120$ feet of fencing to enclose a rectangular plot of land adjacent to a straight river. Knowing that he does not need to fence the side along the river, write the function that gives the area of the plot in terms of the width $x$ perpendicular to the river. What dimensions maximize the area and what is the maximum area?

State the vertical and horizontal asymptotes of the inverse proportion function $f(x) = \frac{2}{x-3} + 4$.

Renting a charter bus for a school trip costs $\$600$. Write the function $C(x)$ that determines the cost per student as a function of the number of students $x$ attending. If at the last minute $5$ students drop out, write the new cost per student function and find how much each will pay if originally $30$ students were going to go.

Given the inverse proportion function $f(x) = \frac{3x - 1}{x + 2}$, find the expression for its inverse function $f^{-1}(x)$. Prove algebraically that the vertical and horizontal asymptotes are swapped between $f(x)$ and $f^{-1}(x)$.

A bacteria culture doubles every hour. If initially there are $100$ bacteria, write the exponential function $N(t)$ that gives the number of bacteria as a function of time $t$ in hours.

$\$5000$ is invested in a bank account that offers an annual compound interest rate of $4\%$. Write the exponential function that models the accumulated balance after $t$ years and calculate the balance after $5$ years.

The amount of Carbon-14 in a fossil decreases exponentially according to the function $N(t) = N_0 \cdot (0,5)^{\frac{t}{5730}}$, where $N_0$ is the initial amount and $t$ is the time in years. If a bone retains only $12,5\%$ of its original Carbon-14, how many years have passed?

Determine the domain of the logarithmic function $f(x) = \log_3(x - 5)$.

The sound intensity level $L$, in decibels (dB), is calculated using the logarithmic function $L(I) = 10 \log_{10} \left( \frac{I}{I_0} \right)$, where $I$ is the sound intensity and $I_0 = 10^{-12} \text{ W/m}^2$ is the threshold of hearing. Calculate the sound intensity level of a normal conversation whose intensity is $I = 10^{-6} \text{ W/m}^2$.

Given the piecewise function $f(x) = \begin{cases} 2x+1 & \text{if } x < 0 \\ x^2 & \text{if } x \ge 0 \end{cases}$, evaluate $f(-2)$, $f(0)$, and $f(3)$.

The following graph shows the speed of a cyclist as a function of time. Determine the algebraic expression of the piecewise function that represents it.

The monthly cost of the water bill in a town is calculated using a piecewise function: a fixed charge of $\$10$ which includes the first $15 \text{ ft}^3$, a cost of $\$0,5$ per additional $\text{ft}^3$ for consumption between $15 \text{ ft}^3$ and $30 \text{ ft}^3$, and a cost of $\$1,2$ for each $\text{ft}^3$ exceeding $30 \text{ ft}^3$. Write the algebraic expression of the cost function $C(x)$ in terms of the total cubic feet consumed $x$.

Express the function $f(x) = |x^2 - 5x + 6|$ as a piecewise function without using absolute value. Indicate in which intervals the original quadratic function was negative before taking the absolute value.

The height $h(t)$ in feet of a Ferris wheel car above the ground as a function of time $t$ in minutes is given by $h(t) = 12 + 10\sin\left(\frac{\pi}{4}t\right)$. What is the maximum and minimum height reached by the Ferris wheel? How long does it take to complete one full revolution?

The trajectory of a projectile describes a parabola with equation $y = ax^2 + bx + c$. Knowing that the projectile is launched from the point $(0, 3)$, reaches its maximum height at the point $(2, 7)$, and falls to the ground, find the equation of the parabola and determine at what horizontal distance from the launch point it will impact.

The graph shows the intersection between the line $y = -x + 5$ and the hyperbola $y = \frac{4}{x}$. Find the exact coordinates of the points of intersection $A$ and $B$.

Determine the values of the parameters $a$, $b$, and $c$ of the rational function $f(x) = \frac{ax + 2}{bx + c}$, knowing that it has a vertical asymptote at $x = 1$, a horizontal asymptote at $y = 3$, and passes through the point $(2, 8)$.