Composition of functions

Given the functions $f(x) = 3x + 2$ and $g(x) = x^2 - 1$, calculate the numerical value of the composite function $(g \circ f)(2)$.

Consider the functions $f(x) = x - 2$, $g(x) = 3x$ and $h(x) = x^2$. Calculate the numerical value of the repeated composite function $(h \circ g \circ f)(4)$.

Aoife has to calculate the numerical value of the composition $(f \circ g)(2)$ for the functions $f(x) = 2x^2 - 1$ and $g(x) = 3x - 5$. On her exam, she writes the following procedure:

"Since I am asked for $(f \circ g)(2)$, I first calculate $f(2) = 2 \times 2^2 - 1 = 7$. Then I enter this result into the second function: $g(7) = 3 \times 7 - 5 = 16$. Therefore, $(f \circ g)(2) = 16$".

1. a) Identify the conceptual error Aoife made and explain, using the theoretical definition of function composition, why her reasoning is not correct.
2. b) Perform the calculations in the proper order to find the correct value of $(f \circ g)(2)$.
3. c) What expression or value did Aoife actually calculate? Justify your answer.

Observe the following graph where the functions $f(x)$ (in blue) and $g(x)$ (in green) are represented. From the visual information and without needing to obtain their algebraic expressions, determine the numerical values of $(f \circ g)(2)$ and $(g \circ f)(-1)$.

Find the algebraic expression for the composite function $(f \circ g)(x)$ given that $f(x) = 5x - 4$ and $g(x) = x^2 + 3x$.

Given the rational functions $f(x) = \frac{2}{x}$ and $g(x) = x + 3$, determine the simplified expression for $(f \circ g)(x)$ and for $(g \circ f)(x)$.

Given the composite function $h(x) = \sqrt{2x + 5}$, express it as the composite of two simpler functions $(g \circ f)(x)$, clearly stating the expression for $f(x)$ and for $g(x)$.

When a stone is dropped into a pond, a circular ripple is formed whose radius increases over time according to the function $r(t) = 2.5t$ (with $r$ in centimetres and $t$ in seconds). Given that the area of the circle is given by $A(r) = \pi r^2$, find the function $(A \circ r)(t)$ describing the area as a function of time and calculate the area of the affected surface after $4$ seconds.

Given that $f(x) = 3x + 2$ and that the composite function is $(g \circ f)(x) = 9x^2 + 12x + 5$, determine the algebraic expression for the outer function $g(x)$.

We know the first-degree polynomial function $f(x) = 2x - 3$, but we do not know the expression of a second function $g(x)$. However, it is known that when they are composed, the resulting quadratic function is:

1. a) Reasonably find the algebraic expression for $g(x)$. To do so, make the change of variable $u = 2x - 3$, solve for $x$ in terms of $u$, and substitute it into the composite expression $(g \circ f)(x)$.
2. b) Check that your deduction is correct by recalculating $g(f(x))$ using the formula you found for $g(x)$.
3. c) Reason whether the function $g(x)$ found is unique or if other distinct functions could exist that produce exactly the same composition with $f(x)$.

Decompose the complex function $H(x) = \frac{1}{\sqrt{3^{\sin x} + 2}}$ as the successive composition of four elementary functions $f_1, f_2, f_3, f_4$ such that $H(x) = (f_4 \circ f_3 \circ f_2 \circ f_1)(x)$. Clearly state the algebraic expression for each of the four functions.

Consider the rational function $f(x) = \frac{1}{x - 3}$ and an unknown function $g(x)$. It is known that the composition of both gives the result:

1. a) Determine, with justification, the algebraic expression of the function $g(x)$.
2. b) Find the domain of the composite function $(f \circ g)(x)$ by strictly applying the formal definition of a chain of functions: $x \in \operatorname{Dom}(g)$ and $g(x) \in \operatorname{Dom}(f)$.
3. c) A student looks at the final expression $\frac{x + 1}{2 - 3x}$ and states: «The domain of $(f \circ g)$ is all real numbers except $x = \frac{2}{3}$, since for $x = -1$ the formula gives $\frac{-1+1}{2-3(-1)} = \frac{0}{5} = 0$, which is a perfectly valid real value». Think of the composition as two chained machines: the first is $g$, which receives the number $x$ and outputs $g(x)$; that output goes directly into the second machine, $f$, which outputs $f(g(x))$. Each machine only works with inputs from its own domain, and if the first one gives no output, the second receives nothing. Using this idea of the two chained machines, refute the student's argument and explain exactly what happens in the calculation process when you try to evaluate the composition at $x = -1$.

Using the functions $f(x) = 5x - 4$ and $g(x) = x^2 + 3x$, find the expression for $(g \circ f)(x)$ and prove analytically that function composition is not commutative.

Prove analytically that composing the function $f(x) = \frac{3x + 1}{x - 2}$ with the identity function $I(x) = x$ yields the original function itself. Calculate and simplify both $(f \circ I)(x)$ and $(I \circ f)(x)$.

An electronics shop applies a discount of $15\%$ to all its products, which is modelled by the function $d(x) = 0.85x$. In addition, a customer has a loyalty voucher for a direct discount of $20$ euros, modelled by the function $c(x) = x - 20$. Determine the expressions for $(c \circ d)(x)$ and $(d \circ c)(x)$. Which order of composition benefits the customer more if they want to buy a computer with an initial price of $600$ euros?

Given the functions $f(x) = 2x + k$ and $g(x) = 4x - 3$, find the value of the parameter $k$ so that the commutative property holds in this particular case, that is, $(f \circ g)(x) = (g \circ f)(x)$ for any value of $x$.

Let $f(x) = \frac{x}{x - 1}$ be the rational function. Calculate the successive compositions $(f \circ f)(x)$ and $(f \circ f \circ f)(x)$. From the results obtained, deduce with justification the expression of $f^{[n]}(x)$ (the composition of $f$ with itself $n$ times) depending on whether $n$ is an even or an odd number.

Prove algebraically that, if $f(x) = \frac{ax + b}{cx - a}$ (with the condition $a^2 + bc \neq 0$), the function is its own inverse by verifying that $(f \circ f)(x) = x$ for all $x$ belonging to its domain.

A light beam passes through a series of optical filters. A type A filter reduces light intensity by $20\%$, modelled by $f_A(I) = 0.8I$. A type B filter reduces intensity by a fixed amount of $15 \text{ W/m}^2$, modelled by $f_B(I) = I - 15$. If the initial intensity is $I_0$, find the expression for the final intensity after passing through $3$ type A filters followed by $2$ type B filters. Is the result equivalent to passing through the $2$ type B filters first and then the $3$ type A filters?

Rigorously determine the domain of the composite function $(g \circ f)(x)$ given that $f(x) = x - 4$ and $g(x) = \sqrt{x}$.

Given the functions $f(x)$ and $g(x)$, where $f(x) = 2x$ if $x < 1$ and $f(x) = x^2$ if $x \geq 1$, and the function $g(x) = x + 3$. Determine the complete algebraic expression for the composite function $(f \circ g)(x)$, paying special attention to the intervals of definition.

Consider the functions $f(x) = \sqrt{x + 3}$ and $g(x) = \frac{1}{x^2 - 4}$. Perform a rigorous analytical study to determine the domain of definition of the composite function $(f \circ g)(x)$.

Observe the graph of the function $f(x)$. From it, obtain the piecewise analytical expression for $f(x)$ and calculate the detailed algebraic expression for $(f \circ f)(x)$.

Given the functions $f(x) = x^2 - 4$ and $g(x) = \frac{1}{\sqrt{x}}$, the composite function $h(x) = (g \circ f)(x)$ is defined.

1. a) Estimate: Before performing any formal algebraic calculation, reflect on the chain of processes: what kind of real values does the function $g$ accept as input to avoid undefined expressions in the real numbers? Keeping this in mind, qualitatively deduce what sign the output of $f(x)$ must have and estimate which intervals of the real line will form the domain of $h(x)$. In particular, reason without computing whether the values $x = 0$ and $x = 2$ can be part of this domain.
2. b) Calculate: Write the analytic formula for $h(x)$ and determine its domain rigorously by applying the two theoretical conditions: $x \in \operatorname{Dom}(f)$ and $f(x) \in \operatorname{Dom}(g)$. Express the result using intervals.
3. c) Check: Compare the domain obtained analytically with your estimates from part a). Explain in detail why, even though $x = 0$ belongs to the domain of $f$, it is ultimately excluded from the composition.