Functions of a real variable (review)

Determine the domain of the following functions of a real variable based on the restrictions of the operations that define them:

1. a) $f(x) = \frac{2x - 5}{x^2 - 9}$

1. b) $g(x) = \sqrt{3x + 12}$

1. c) $h(x) = \ln(x - 1)$

Classify the following functions according to their type (polynomial, rational, radical, exponential, or logarithmic):

1. a) $f(x) = 3^x - 1$

1. b) $g(x) = \frac{x^2 + 1}{x - 2}$

1. c) $h(x) = \log_{10}(x^2)$

1. d) $i(x) = x^3 - 5x + 2$

1. e) $j(x) = \sqrt[3]{x - 4}$

A phone company offers a data plan where the first $10$ GB cost a fixed $15$ €. From then on, each extra GB is charged at $2$ €. The function representing the total cost $C(x)$ in terms of the GB consumed $x$ is:

$C(x) = \begin{cases} 15 & \text{if } 0 \le x \le 10 \\ 15 + 2(x - 10) & \text{if } x > 10 \end{cases}$

Study whether the function is continuous at the critical point $x = 10$.

Given the modulus function $f(x) = |x - 3|$, express it as a piecewise function and calculate its value for $x = 0$ and $x = 5$.

A new video game streaming service applies a monthly fee based on high-speed data usage. The first $20$ GB are included in a fixed charge of $15$ €. Beyond that, the price for each additional GB increases linearly according to the function $P(x) = 15 + 0,5(x - 20)$, where $x$ is the total number of GB consumed.

However, if the user exceeds $100$ GB, the company applies an excess traffic penalty and the function becomes $P(x) = 0,1x^2 - 45$.

Write the algebraic expression for the total cost function $C(x)$ as a piecewise-defined function and check whether the price "jumps" or is continuous right at the point of consuming $100$ GB.

Given the functions $f(x) = x + 3$ and $g(x) = x^2 - 1$, perform the following operations and determine the domain of the resulting function:

1. a) $(f + g)(x)$

1. b) $(f \cdot g)(x)$

1. c) $(\frac{f}{g})(x)$

Consider the functions $f(x) = \sqrt{x + 2}$ and $g(x) = \frac{1}{x^2 - 1}$.

1. Find the expression for the composite function $(g \circ f)(x)$.

2. Determine the domain of $(g \circ f)(x)$.

3. Does this domain coincide with that of the function $h(x) = \frac{1}{x+1}$? Justify your answer.

Find the inverse function $f^{-1}(x)$ of the function $f(x) = \frac{e^x + 3}{2}$. State the domain and range of both functions and verify graphically their symmetry with respect to the line y = x.

Study the symmetry (even, odd or neither) of the following functions, justifying the steps by evaluating $f(-x)$:

1. $f(x) = \frac{x^3}{x^2 + 1}$

2. $g(x) = x \cdot \cos(x)$

3. $h(x) = \frac{e^x + e^{-x}}{2}$ (Hyperbolic cosine function)

Consider the family of functions $f_k(x) = \frac{x^2 - k^2}{x^2 + k^2}$ where $k \in \mathbb{R}, k \neq 0$.

1. Show that for any value of $k$, the function is even.

2. Determine the axis intercepts in terms of $k$.

3. Find the value of $k$ such that the function takes the value $y = 1/2$ when $x = 3$.

4. Discuss whether there is any value of $k$ for which the function has vertical asymptotes.

Any function $f(x)$ can be uniquely decomposed, at the points $x$ where both $f(x)$ and $f(-x)$ exist, as the sum of an even function $g(x)$ and an odd function $h(x)$, such that:

Given the function $f(x) = \frac{x+2}{x-1}$, find its even and odd components. Verify algebraically that $g(x)$ is indeed an even function and $h(x)$ is an odd function. What are the domains of each of these three functions?