Composition of Functions

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

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

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 of the composite function $(f \circ g)(x)$ knowing 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 constant function $f(x) = 7$ and the quadratic function $g(x) = 2x^2 - x$, determine the algebraic expressions for $(f \circ g)(x)$ and $(g \circ f)(x)$.

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 the composition of functions is not commutative.

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

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

When throwing a stone into a pond, a circular wave is formed whose radius increases with time according to the function $r(t) = 2.5t$ (with $r$ in inches and $t$ in seconds). Knowing that the area of the circle is given by $A(r) = \pi r^2$, find the function $(A \circ r)(t)$ that describes the area as a function of time and calculate the area of the affected surface at $4$ seconds.

The population of a bacterium in a culture depends on the temperature according to the function $P(T) = 50T^2 - 100T + 200$. In turn, the temperature of the incubator varies with time in hours according to $T(t) = 1.5t + 10$. Calculate the exact population of bacteria there will be after $6$ hours have elapsed.

The daily production cost of $x$ parts in a factory is given by the function $C(x) = 15x + 500$ dollars. The number of parts assembled by a machine in $t$ hours of continuous operation is $x(t) = 40t$. Express the production cost as a function of the machine's operating time and calculate the total cost after a working day of $8$ hours.

An electronics store applies a $15\%$ discount to all its products, which is modeled by the function $d(x) = 0.85x$. In addition, a customer has a loyalty coupon for a direct discount of $\$20$, modeled 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 most if they want to buy a computer whose initial price is $\$600$?

A spherical hot-air balloon is inflated by a compressor that injects air at a constant rate, so that its volume in cubic feet is given by $V(t) = 12t$ (with $t$ in minutes). Knowing that the volume of a sphere is $V = \frac{4}{3}\pi r^3$ and its surface area is $S = 4\pi r^2$, find the composite function $S(t)$ that gives the surface area of the balloon directly as a function of the elapsed time.

A beam of light passes through a series of optical filters. A type A filter reduces the luminous intensity by $20\%$, modeled by $f_A(I) = 0.8I$. A type B filter reduces the intensity by a fixed amount of $15 \text{ W/m}^2$, modeled 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 first through the $2$ type B filters and then the $3$ type A filters?

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

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

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

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

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 indicate the algebraic expression of each of the four functions.

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 analytical expression of the composite function $(f \circ g)(x)$, paying special attention to the intervals of definition.

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

Let the rational function be $f(x) = \frac{x}{x - 1}$. Calculate the successive compositions $(f \circ f)(x)$ and $(f \circ f \circ f)(x)$. From the results obtained, logically deduce what the expression for $f^{[n]}(x)$ (the composition of $f$ with itself $n$ times) will be, depending on whether $n$ is an even or 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 checking that $(f \circ f)(x) = x$ for all $x$ belonging to its domain.