Representing Functions

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

The time $t$ (in seconds) taken for a dropped object to fall to the ground from a height $h$ (in meters) is approximated by the radical function $t(h) = \sqrt{\frac{2h}{9.8}}$. Find the domain of the function in this context and determine the time taken for an object dropped from a height of $4.9$ meters to fall.

A taxi driver charges a base fare of $60$ pesos and $30$ pesos for each kilometer traveled. Write the algebraic expression for the function relating the cost of the trip to the distance traveled in kilometers.

A line passes through the points $(-3, -4)$ and $(0, 4)$. 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 $f(x) = x^2 - 6x + 5$.

Starting from the base parabola $y = x^2$ (dashed line), a translation has been applied to obtain the parabola shown as a solid line. Write down the algebraic expression of the new parabola and state the translation vector applied.

The profit from selling end-of-year party T-shirts, in pesos, is given by the function $B(x) = -40x^2 + 2\,400x - 20\,000$, where $x$ is the number of T-shirts sold in hundreds. How many T-shirts must be sold to maximize profit, and what is this maximum profit?

A farmer has $120$ meters of fencing to enclose a rectangular plot of land adjacent to a straight river. Given that no fencing is needed along the riverbank, write down the function that gives the area of the plot as a function 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 $f(x) = \frac{2}{x-3} + 4$.

Hiring a coach for a school trip costs $12\,000$ pesos. Write down the function $C(x)$ that gives the cost per student depending on the number of students $x$ attending. If $5$ students cancel at the last minute, write the new cost-per-student function and find how much each student will pay if $30$ students were originally going.

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

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

In a quiz game, the points accumulated double after each completed round. If you start with $100$ points, write down the exponential function $P(r)$ that gives the points as a function of the number of completed rounds $r$.

An amount of $100\,000$ pesos is invested in a bank deposit offering an annual compound interest rate of $4\%$. Write the exponential function that models the accumulated capital after $t$ years and calculate the balance available after $5$ years.

The amount of Carbon-14 in a fossil remnant decays exponentially according to the function $N(t) = N_0 \times (0,5)^{\frac{t}{5730}}$, 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 with an intensity of $I = 10^{-6} \text{ W/m}^2$.

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

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

The monthly water bill in a municipality is calculated using a piecewise function: a base fee of $200$ pesos including the first $15 \text{ m}^3$, a cost of $10$ pesos per additional $\text{m}^3$ for consumption between $15 \text{ m}^3$ and $30 \text{ m}^3$, and a cost of $24$ pesos for each $\text{m}^3$ exceeding $30 \text{ m}^3$. Write the algebraic expression for the cost function $C(x)$ in terms of the total cubic meters consumed $x$.

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

The height $h(t)$ in meters of a Ferris wheel cabin 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 are the maximum and minimum heights reached by the Ferris wheel? How long does it take to complete one full revolution?

The trajectory of a projectile follows a parabola with equation $y = ax^2 + bx + c$. Given that the projectile is launched from point $(0, 3)$, reaches its maximum height at point $(2, 7)$, and hits the ground, find the algebraic expression of the parabola and determine the horizontal distance from the launch point at which it lands.

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