Characteristics of Functions

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

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

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

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

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

Find the domain of the function $f(x) = \sqrt{\log_2(x-3)}$, justifying your answer.

Find the axis intercepts of the polynomial function $h(x) = x^2 - 5x + 6$.

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

Find the axis intercepts and study the sign of the rational function $g(x) = \frac{x^2-4}{x-1}$.

Given the piecewise function:

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

Find its axis intercepts and study its sign across its whole domain.

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

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

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

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

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

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.

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

A farmer has $100$ meters of fencing to enclose a rectangular vegetable garden taking advantage of an existing stone wall (no fencing is needed on that side). If $x$ is the length of the sides perpendicular to the wall, the function describing 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.

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 pesos) per unit produced in an electronic components factory is given by the function $C_m(x) = \frac{6\,000x + 24\,000}{x}$, 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.

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

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

The attached graph shows the height $h(t)$ in meters of a Ferris wheel capsule as a function of time $t$ in minutes. Given that the motion is modeled by a trigonometric function, determine its period, the maximum and minimum heights reached by the capsule, and the times at which they are reached during the first full cycle.

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