Exercises: Characteristics of functions

Determine the domain of definition of the rational function $f(x) = \frac{x-1}{x^2-9}$.

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

Find the points of intersection with the coordinate axes of the polynomial function $h(x) = x^2 - 5x + 6$.

Study the symmetry (even, odd or neither) of the polynomial function $p(x) = x^4 - 2x^2 + 7$.

Determine the intervals in which the function $q(x) = -3x + 12$ has a positive sign and in which it has a negative sign.

State the range of the quadratic function $r(x) = x^2 + 4$.

From the graphical representation of the parabola $y = f(x)$, state its intervals of increase and decrease, as well as the coordinates of its relative extremum.

Calculate the domain of the logarithmic function $s(x) = \log_3(x+2)$.

Find the domain of definition of the function $f(x) = \sqrt{\frac{x+1}{x-3}}$.

Calculate the points of intersection with the axes and study the sign of the rational function $g(x) = \frac{x^2-4}{x-1}$.

Determine analytically whether the function $h(x) = \frac{x^3}{x^2+1}$ is even, odd or has no symmetry.

A projectile is launched vertically upward. The height $h(t)$ in feet it reaches after $t$ seconds is given by the function $h(t) = -5t^2 + 20t + 1$. Determine at what instant it reaches the maximum height and what the value of this height is.

A farmer has $100$ feet of fencing to enclose a rectangular vegetable patch, taking advantage of an existing stone wall (no fence is needed on that side). If $x$ is the length of the sides perpendicular to the wall, the function that describes the area is $A(x) = x(100-2x)$. Determine the domain of the function in the context of the problem and calculate the dimensions that maximize the area.

Observe the graph of the following periodic function that models the voltage in an electronic circuit. Determine its period $T$ and calculate the value of $f(17)$.

Given the piecewise-defined function:

$f(x) = \begin{cases} 2x+6 & \text{if } x < -1 \\ x^2+3 & \text{if } x \ge -1 \end{cases}$

Find its points of intersection with the axes and study its sign over its entire domain.

Determine the domain of definition of the function $f(x) = \ln(x^2 - 4x + 3)$.

Calculate the domain of the function $f(x) = \sqrt{\log_2(x-3)}$, showing your reasoning.

Determine the coefficients $a$, $b$ and $c$ of the quadratic function $f(x) = ax^2 + bx + c$ knowing that its vertex is the point $V(3, -4)$ and that it intersects the x-axis at $x = 1$.

The average monthly cost (in dollars) per unit produced in an electronic components factory is given by the function $C_m(x) = \frac{300x + 1200}{x}$, where $x$ is the number of units produced ($x > 0$). Determine the domain in the context of the problem, study its sign and prove analytically in which intervals the average cost is increasing or decreasing.

Prove analytically that the function $f(x) = \ln\left(\frac{1-x}{1+x}\right)$ is odd and determine its domain of definition.

The attached graph shows the height $h(t)$ in feet of a Ferris wheel cabin as a function of time $t$ in minutes. Knowing that the movement is modeled by a trigonometric function, determine its period, the maximum and minimum heights reached by the cabin, and the instants of time at which they are reached during the first complete cycle.

Find the values of the parameters $a$ and $b$ so that the function $f(x) = \frac{x^2+a}{x-b}$ has the domain $\mathbb{R} \setminus \{2\}$ and intersects the x-axis at the points $x = 4$ and $x = -4$.

Calculate the inverse function of $f(x) = \frac{2x-5}{x+1}$ and use it to determine the range of the function $f(x)$.

From the graph of the function $y = f(x)$, carry out a complete analysis indicating: domain, range, symmetry, equations of the asymptotes, relative extrema and intervals of positive and negative sign.