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 euros) is given by the function $I(x) = 15x$, where $x$ is the number of packages sent. On the other hand, its total monthly costs (in euros) follow the function $C(x) = 5x + 1200$.

Obtain the expression of 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 $(f \cdot g)(x)$ 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) $(f \circ f)(0)$

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.

In a Mathematics Applied to the Social Sciences exam, Marcos must perform operations with the functions $f(x) = 3x^2 - 2x + 5$ and $g(x) = x - 4$. On his answer sheet, he writes the following:

1. a) $(f - g)(x) = 3x^2 - 2x + 5 - x - 4 = 3x^2 - 3x + 1$.
2. b) $(g \circ f)(2) = g(2) \cdot f(2) = (2 - 4) \cdot (3\cdot 2^2 - 2\cdot 2 + 5) = (-2) \cdot 13 = -26$.

Reasonably identify the two conceptual errors that Marcos has made: one in the subtraction of functions and another when interpreting the notation of function composition. Correct both parts by calculating the simplified and correct algebraic expression of $(f - g)(x)$ and the exact numerical value of $(g \circ f)(2)$.

During the sale period in a clothing store in Madrid, two successive promotions applicable to a garment with a tag price of $x$ euros are announced:

• Promotion A: a direct $20\,\%$ discount on the price, modeled by the function $f(x) = 0,80x$.

• Promotion B: an additional discount coupon of $15$ € applicable to the purchase, modeled by the function $g(x) = x - 15$.

• Model algebraically through composition the function representing the final price if the $20\,\%$ discount is applied first and then the $15$ € coupon, that is, $(g \circ f)(x)$.

• Model the function if the store decided to apply the $15$ € coupon first and, on that remaining amount, the $20\,\%$ discount, that is, $(f \circ g)(x)$.

• Compare both functions by calculating their difference: $[(g \circ f) - (f \circ g)](x)$. Based on the result, determine which of the two orders of application benefits the buyer more and how much exact money they save regardless of the garment's initial price $x$.

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 $f(x) = \frac{3x - 1}{x + 2}$ 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 $2$ € is charged for each purchase, represented by $c(x) = x + 2$. In both cases, $x$ is the base price of the product, in euros.

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 metres 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 $f(x) = \dfrac{x + 4}{x - 2}$ and $g(x) = \sqrt{x - 1}$, the quotient function is defined as $H(x) = \left(\dfrac{f}{g}\right)(x)$.

1. Before carrying out any algebraic calculations, reasonably estimate the sign of $H(5)$ by analyzing the sign of $f(5)$ and that of $g(5)$ separately. Also estimate whether $H(5)$ will be greater or less than $2$ by considering the orders of magnitude of the numerator and the denominator. Does it make sense to try to calculate $H(0)$ or $H(2)$? Justify your prior estimation regarding the feasibility of those calculations based on the restrictions of $f$ and $g$.

2. Obtain the simplified algebraic expression of $H(x)$ and rigorously determine its domain of definition $\text{Dom}(H)$.

3. Calculate the exact numerical value of $H(5)$ expressing it with radicals, obtain its approximation to two decimal places, and check whether your initial estimation regarding its sign and size was correct.

Consider the linear function $f(x) = 2x - 5$. We are looking for a function $g(x)$ such that, when composed with $f$, the result is the quadratic function $(g \circ f)(x) = 4x^2 - 20x + 21$.

1. By means of a change of variable of the type $t = 2x - 5$ (or by expressing the quadratic polynomial in terms of the binomial $2x - 5$), reasonably deduce the algebraic expression of the unknown function $g(t)$.

2. Verify that your deduction is correct by evaluating step by step the composition $(g \circ f)(x)$ from the formula found.

3. Find the composition in the reverse order, $(f \circ g)(x)$, and compare it with $(g \circ f)(x)$. Does the commutative property hold in this case? Justify your answer by indicating whether there is any particular value of $x$ for which both compositions coincide.

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 a flight path whose height is defined by the function $h(t) = -t^2 + 10t - 16$, where $h$ is the height in tens of meters and $t$ is the time in minutes since the sensor is activated, with $t \in [2, 8]$. The figure shows the graph of the path.

1. a) Explain why the function $h(t)$ has no inverse on its whole domain $[2, 8]$.

1. b) Restrict the domain of $h(t)$ to the interval where the drone is descending so that the function is injective.

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

A city centre car park 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 points of inflection or changes in gradient.

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 behaviour before and after $x = 5$.

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

In the stock market, the price of a share $P$ (in euros) 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 euros) 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 $20$ euros.

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 $f(x) = 3 \cdot 2^{x-1} + 4$.

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 $R(x) = 500x - 2x^2$ and the cost function is $C(x) = 100x + 3000$, 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 maximise 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 $P(t) = \frac{100}{1 + 9 \cdot e^{-0,5t}}$. 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).

Consider the functions $f(x) = \dfrac{x + 1}{x - 3}$ and $g(x) = \dfrac{2}{x - 1}$. To find the domain of the composite function $(g \circ f)(x)$, two 1st-year Bachillerato students argue about which of the following methods is correct:

• Method 1 (Álvaro): First calculates the simplified formula of $(g \circ f)(x)$ and then looks for the values that make the resulting denominator zero:

$$(g \circ f)(x) = \dfrac{2}{\frac{x+1}{x-3} - 1} = \dfrac{2}{\frac{x+1 - (x-3)}{x-3}} = \dfrac{2(x-3)}{4} = \dfrac{x-3}{2}$$

Since the final result is a degree-1 polynomial, Álvaro concludes that $\text{Dom}(g \circ f) = \mathbb{R}$.

• Method 2 (Elena): Applies the formal definition of the domain of a composition: simultaneously requires that $x \in \text{Dom}(f)$ and that the intermediate output $f(x) \in \text{Dom}(g)$.

• Analyze Álvaro's procedure and explain with mathematical rigor why his method leads to a serious error. What structural information is lost when simplifying the algebraic fraction?

• Work through Method 2 proposed by Elena step by step and determine the real and complete domain of $(g \circ f)(x)$.

• Imagine the process as an assembly line and explain what exactly happens when evaluating the chain of operations for the inputs $x = 3$ and $x = 1$. Conclude which of the two methods should always be chosen when studying composite functions and state a general guideline to avoid Álvaro's error.

An agricultural cooperative in Jaén markets extra virgin olive oil in special containers. Its economic management team models its activity using the following functions:

• The number of containers produced per month, $q$, depends on the number of workers hired at the plant, $x$, according to the function $q(x) = 80x$.

• The monthly production costs in euros are given by $C(x) = 1400x + 3600$, which includes wages and fixed infrastructure.

• The unit selling price per container $p$ (in euros) depends on the total volume of containers launched onto the market to avoid surplus stock, according to the demand function $p(q) = 28 - 0,00625q$.

• Determine the cooperative's monthly revenue function in terms of the number of containers produced, $I(q) = q \cdot p(q)$, using the product of functions.

• Model the direct monthly revenue function in terms of the number of workers, $(I \circ q)(x)$, through function composition. Justify what this composition represents in the company's management.

• Obtain the net monthly profit function $B(x) = (I \circ q)(x) - C(x)$ by subtracting the cost function from the revenue function. Determine how many workers the cooperative must hire to reach the break-even point ($B(x) = 0$), solve the resulting equation, and interpret which hiring interval guarantees that the cooperative earns strictly positive profits.