Exercises: Concept of a function

Given the real function of a real variable $f(x) = -2x^2 + 4x - 1$, calculate the images corresponding to the values $x = -2$, $x = 0$ and $x = 3$.

Observe the two curves represented in panels A and B. Determine, justifying your answer using the concept of a real function of a real variable, which of the two graphs corresponds to the representation of a function $y = f(x)$ and which does not.

A plumber charges $\$35$ for the service call fee, plus $\$22$ for each hour of labor. Write the analytical expression of the function $C(t)$ that determines the total cost of the repair in terms of the time $t$ (in hours) spent.

Indicate, reasoning your answer, whether the following relationships between quantities define a mathematical function $y = f(x)$:

• Each integer $x$ is associated with its cube $y$.

• Each positive real number $x$ is associated with its square roots $y$.

Given the following table of values relating two variables $x$ and $y$, justify whether it can correspond to a mathematical function $y = f(x)$.

• Values of $x$: $-2, -1, 0, 1, -1, 3$

• Values of $y$: $4, 1, 0, 1, 2, 9$

Given the function $f(x) = 3x - 7$, find the pre-image (or pre-images) of the value $y = 8$. That is, find the value or values of $x$ such that $f(x) = 8$.

Calculate the numerical value of the function $g(x) = \frac{x^2 - 1}{2x + 3}$ for $x = \frac{1}{2}$ and for $x = -1$.

Identify the independent variable and the dependent variable in the following everyday situations:

• The price paid for a piece of cheese and its weight on the scales.

• The water level of a tank that is being emptied and the time elapsed since the drain valve was opened.

Express analytically the area $A(x)$ of a rectangular enclosure whose perimeter is $40$ ft, in terms of the length $x$ of its base. Also indicate what the logical domain of this function should be, considering the geometric context of the problem.

The height $h$ (in feet) of a projectile launched vertically upwards from a platform is given by the function $h(t) = -5t^2 + 30t + 10$, where $t$ is the time elapsed in seconds.

Calculate the initial height from which it was launched, the height at $2$ seconds, and set up the equation you would solve to find out at what instant the projectile hits the ground.

A company manufactures electronic components. The fixed monthly maintenance costs of the warehouse amount to $\$1500$, and manufacturing each component has an additional cost of $\$12$. If each component is sold for $\$25$, determine the Profit function $B(x)$ in terms of the number $x$ of components manufactured and sold per month. How many components must be sold to start making a profit?

It is known that the graph of a quadratic function of the form $f(x) = ax^2 + bx + c$ passes through the points $(0, -3)$, $(1, -4)$ and $(-1, 0)$. Set up and solve the corresponding system of equations to determine the values of the parameters $a$, $b$ and $c$.

The following graph shows the evolution of the temperature $T$ (in $^\circ$°F) of a chemical solution in terms of the time $t$ (in minutes) elapsed since the start of a reaction. From the graph, determine the initial temperature of the solution and the instants of time when the temperature was exactly $20^\circ$°F.

Given the polynomial function $f(x) = x^2 - 4x + 5$, obtain the simplified and expanded expression of $f(x+2) - f(x-1)$.

Consider the mathematical relationship given by the implicit equation $x^2 + y^2 = 16$.

Explain why this equation does not define $y$ as a single function of $x$. Write the analytical expressions of the two functions $f_1(x)$ and $f_2(x)$ that can be extracted by making the variable $y$ the subject.

A mobile network operator offers a roaming data tariff with the following conditions: a fixed fee of $\$5$ is charged, which includes up to $2$ Gigabytes (GB) of usage. From $2$ GB onwards, $\$1.50$ is charged for each additional GB used. Write the analytical expression of the piecewise function $C(x)$ that provides the total cost in terms of the GB used $x$.

A rectangle is inscribed in an isosceles triangle with a base of $12$ in and a height of $8$ in, such that one of the sides of the rectangle rests on the base of the triangle. Express the area of the rectangle $A(x)$ in terms of the length $x$ of its base.

Given the rational function $f(x) = \frac{2}{x-1}$, calculate and fully simplify the expression of the difference quotient:

$ \frac{f(x+h) - f(x)}{h} $

Assume that $h \neq 0$ and that both $x$ and $x+h$ belong to the domain of the function.

The shipping cost of a package by a courier company is $\$4$ for the first pound (or fraction of a pound), plus $\$1.50$ for each additional pound (or fraction).

Express the total shipping cost $C(x)$ in terms of the weight $x$ of the package in pounds (for $x > 0$). You can use the "greatest integer" or "ceiling" function if you consider it necessary, or define it piecewise for the first intervals. Calculate the exact cost of sending a package of $3.2$ lbs.

A rational function has the form $f(x) = \frac{ax+b}{x+c}$. We know that its graph passes through the origin of coordinates, that it is not defined for $x = -2$, and that $f(1) = 3$. Determine the values of the parameters $a$, $b$ and $c$, and write the final analytical expression of the function.

A right circular cylinder is inscribed inside a hollow sphere of radius $R$. Express the volume of the cylinder $V(h)$ exclusively in terms of its height $h$ and the constant parameter $R$. What is the logical domain of this function in the geometric context of the problem?

In a market, the quantity of an item that consumers are willing to buy depends on the price of the item. The demand relationship is given by $q = 2000 - 40p$, where $q$ is the quantity and $p$ is the unit price in dollars.

Knowing that the total revenue is calculated as the product of the unit price and the quantity, express the revenue function $I(q)$ depending exclusively on the quantity demanded $q$.

Given the mathematical relationship $x = y^2 - 6y + 5$, we want to express $y$ as a function of $x$.

Solve the equation to make $y$ the subject in terms of $x$. How many functions do you obtain? Determine the domain of these functions so that the values of $y$ are real numbers.

Two vehicles depart from the same intersection of roads that form a right angle. Vehicle A travels North at a constant speed of $60$ mph, and vehicle B travels East at $80$ mph. If vehicle A leaves at 10:00 and vehicle B leaves at 11:00, express the distance $d(t)$ separating both vehicles as a function of the time $t$ (in hours) elapsed since 10:00, assuming that $t \ge 1$.