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 $(f - g)(x)$.

A courier company determines that its monthly revenue (in pesos) is given by the function $I(x) = 300x$, where $x$ is the number of packages sent. On the other hand, its total costs are governed by the function $C(x) = 100x + 24\,000$.

Obtain the expression for the profit function $B(x) = (I - C)(x)$ and explain what the constant term of the resulting function represents in this context.

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).

A market research study for a new mobile application determines that the monthly revenue function is $R(x) = 10\,000x - 40x^2$ and the cost function is $C(x) = 2\,000x + 60\,000$, where $x$ is the number of premium subscriptions sold.

1. Find the net profit function $B(x) = (R - C)(x)$.

2. Determine the number of subscriptions required to maximize profit using the properties of quadratic functions.

3. For which values of $x$ does the company make a loss? Solve the inequality $(R - C)(x) < 0$.

A downtown parking lot uses two functions to calculate the parking fee. The function $f(x) = |x - 5|$ represents an adjustment for time slot and $g(x) = 2x + 3$ represents the base price, where $x$ is the number of hours parked.

1. Define the sum function $S(x) = (f + g)(x)$ as a piecewise-defined function.

2. Sketch the graph of $S(x)$ and identify its inflection points or changes in slope.

3. If a driver leaves their car for $x = 2$ hours, what is the total price? What if they leave it for $x = 10$ hours? Comment on the difference in the function's behavior before and after $x = 5$.

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

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

Calculate the simplified expression of the Quotient function $Q(x) = (f / g)(x)$ and determine its domain of definition, taking into account the restrictions of the original functions and of the operation itself.

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) $(f \circ f)(0)$

In a biology laboratory, the number of individuals in an insect population as a function of time $t$ (in days) follows the function $N(t) = 3t + 100$. It has been observed that the area occupied by this population (in $m^2$) depends on the number of individuals according to the function $A(n) = \sqrt{n}$.

Find the function $(A \circ N)(t)$ that expresses the area occupied as a function of time, and calculate how many square meters they will occupy after 25 days.

An online shop applies a $15\%$ discount to the price of its products, which can be expressed as $d(x) = 0.85x$. In addition, a fixed handling fee of $2$ pesos is charged for each purchase, represented by $c(x) = x + 2$. In both cases, $x$ is the base price of the product, in pesos.

The diagram shows the order in which the functions are applied in each option: in Option A the discount is applied first and then the fee; in Option B, the other way round.

1. a) Find the expression for $(c \circ d)(x)$ and explain what it represents in the context of the problem.
2. b) Find the expression for $(d \circ c)(x)$ and explain whether the final price is the same as in the previous part.

Find the algebraic expression of the composite function $(g \circ f)(x)$ where $f(x) = 3x + 2$ and $g(x) = \sqrt{x - 1}$. State the domain of the resulting function.

The figure shows the diagram of the composition: the value $x$ enters $f$ and the result that comes out of $f$ enters $g$.

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.

Consider the function $f(x) = \sqrt{2x - 6}$.

1. a) Calculate its inverse function $f^{-1}(x)$.
2. b) Determine the domain and range of both $f(x)$ and $f^{-1}(x)$ and check the relationship between them.

Find the inverse function of $f(x) = \frac{2x + 5}{x - 3}$. Once obtained, carry out the following checks to verify that your calculation is correct:

1. a) Calculate $(f \circ f^{-1})(x)$.
2. b) Check that the point $(4, 13)$ belongs to the graph of $f$ and that the point $(13, 4)$ belongs to that of $f^{-1}$.

Consider the exponential function $f(x) = 3 \times 2^{x-1} + 4$.

1. Calculate its inverse function $f^{-1}(x)$ by rearranging for the variable $x$ step by step.

2. Determine the domain and range of both $f(x)$ and $f^{-1}(x)$. Explain the relationship between the range of one and the domain of the other.

3. Check algebraically that $(f \circ f^{-1})(x) = x$.

A forest surveillance drone follows an altitude trajectory defined by the function $h(t) = -t^2 + 10t - 16$, where $h$ is the altitude in tens of meters and $t$ is the time in minutes since the sensor is activated, with $t \in [2, 8]$.

1. Explain why the function $h(t)$ does not have an inverse over its entire domain $[2, 8]$.

2. Restrict the domain of $h(t)$ to the interval where the drone is descending so that the function is one-to-one.

3. Calculate the expression for the inverse function $h^{-1}(x)$ for this descent interval. What information does this inverse function provide in the context of the problem?

In the stock market, the price of a share $P$ (in pesos) depends on the confidence index $I$ according to the function $P(I) = \frac{40I}{I + 10}$. In turn, the confidence index depends on the investment in technology $x$ (in millions of pesos) according to $I(x) = \sqrt{x} + 2$.

1. Obtain the function $(P \circ I)(x)$ that directly relates the share price to the investment in technology.

2. Calculate the investment required for the share price to be $20$ pesos.

3. Determine the inverse function $(P \circ I)^{-1}(y)$. What restrictions must $y$ have for this inverse to make physical sense in this market?

Given the piecewise function:

$f(x) = \begin{cases} 2x + 1 & \text{if } x < 0 \\ x^2 - 1 & \text{if } x \geq 0 \end{cases}$

and the linear function $g(x) = x - 2$.

1. Determine the expression for the composite function $(f \circ g)(x)$. Be careful, as the result will also be a piecewise-defined function, and the sub-domains of $f$ will be affected by the translation of $g$.

2. Calculate the x-intercepts of $(f \circ g)(x)$ (with the x-axis, $OX$).

3. Study the continuity of the resulting function $(f \circ g)(x)$ at the point where its definition changes.

In a biology laboratory, the population of a bacterium at time $t$ (hours) is given by the function $P(t) = \frac{100}{1 + 9 \times e^{-0.5t}}$. The graph shows the evolution of $P(t)$ during the first 12 hours.

1. If we call $f(t) = e^{-0.5t}$, express $P(t)$ as a combination of basic operations (addition, product, quotient) between constant functions and the function $f(t)$.

2. Calculate the time $t$ needed for the population to reach 50 bacteria. To do this, you will have to invert the function $P(t)$ algebraically.

3. What is the limit of the population as time tends to infinity ($t \to \infty$)? Interpret the result from the point of view of operations with functions (what happens to the denominator as $t$ grows).