Exercises: 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.

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?

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

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 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.

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

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.

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.

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.

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 $f^{-1}$.

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.

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.

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?

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 $x = 5$.

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?

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 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 $(R - C)(x) < 0$.

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).