Operations with functions

Given the functions $f(x) = 3x^2 - 5x + 2$ and $g(x) = x^2 + 4x - 7$, determine the simplified expression for the sum function $(f + g)(x)$ and the difference function $(f - g)(x)$.

An artisan pastry workshop has two types of monthly costs. The cost of ovens and energy is given by the function $F(x) = 0,5x^2 + 200$, where $x$ is the number of cakes made. On the other hand, the costs of ingredients and staff follow the function $V(x) = 30x + 500$.

Calculate the expression of the Total Cost function $C(x)$ as the sum of both and determine the cost for the workshop of making 100 cakes in a month.

A young entrepreneur sells designer t-shirts. Their revenue from selling $x$ units is given by the function $I(x) = 25x$, while the production costs are $C(x) = x^2 + 5x + 40$.

Find the algebraic expression of the Profit function, defined as $B(x) = (I - C)(x)$. State how many t-shirts they must sell for the profit to be zero (break-even point).

Given the functions $f(x) = \frac{2x + 1}{x - 3}$ and $g(x) = x + 5$, calculate the domain of the product function $(f \cdot g)(x)$ and the quotient function $(\frac{f}{g})(x)$. Which values of $x$ must be excluded in each case?

Consider the functions $f(x) = 2x - 3$ and $g(x) = x^2 + 1$. Calculate the value of the following composites at the given points:

1. a) $(g \circ f)(2)$
2. b) $(f \circ g)(-1)$
3. c) $(f \circ f)(0)$

Given the functions $f(x) = \frac{3x - 1}{x + 2}$ and $g(x) = x^2 + 1$.

Obtain the expression for the composite function $(f \circ g)(x)$ and calculate its value for $x = 2$. Does the value $x = -2$ belong to the domain of this new function? Justify your answer.

Given the functions $f(x) = \sqrt{x + 3}$ and $g(x) = \frac{1}{x^2 - 4}$, you are asked to carry out a thorough study of their composition. The figure shows the graph of $f$.

1. Determine the algebraic expression of the composite function $(g \circ f)(x)$.

2. Calculate, with justification, the domain of definition of $(g \circ f)(x)$. Keep in mind that for the composition to exist, $x$ must belong to the domain of $f$ and $f(x)$ must belong to the domain of $g$.

3. Is it possible to calculate $(f \circ g)(0)$? Justify your answer by analyzing the domains of both functions.

Given the function $f(x) = \frac{x - 4}{2}$, calculate its inverse function $f^{-1}(x)$. Check that your result is correct by performing the composite $(f \circ f^{-1})(x)$ and verifying that you obtain the identity function.

In a market study, the supply of a product depends on the price $p$ according to the function $O(p) = 2p - 10$. Find the inverse function $O^{-1}$, which gives the price required to obtain a given amount of supply. What would the price be if we want the supply to be 40 units?

Given the graph of the function $f(x) = x + 3$, sketch its inverse function $f^{-1}(x)$ using the symmetry with respect to the line $y = x$. Write the analytic expression of both functions.

The figure shows the graph of $f$ (in blue) and the line $y = x$ (dashed, in orange). Each tick mark on the axes corresponds to one unit.

You can do it on paper and send a photo with your answer.