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 $(h \circ g \circ f)(4)$.

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

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 centimeters 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.

The daily production cost of $x$ parts in a factory is given by the function $C(x) = 300x + 10\,000$ pesos. 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 shift of $8$ hours.

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.

An electronics store applies a discount of $15\%$ to all its products, which is modeled by the function $d(x) = 0.85x$. In addition, a customer has a loyalty voucher for a direct discount of $400$ pesos, modeled by the function $c(x) = x - 400$. 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 $12\,000$ pesos?

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 $x$.

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

Consider the functions $f(x) = \sqrt{x + 3}$ and $g(x) = \frac{1}{x^2 - 4}$. Perform 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 composite of two simpler functions $(g \circ f)(x)$, clearly stating the expression for $f(x)$ and for $g(x)$.

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 $g(x)$.

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.

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.

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

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 $(I \circ f)(x)$.

The electricity demand of a city, in megawatts, depends on the ambient temperature according to the function $D(T) = 50T^2 - 100T + 200$. In turn, the temperature varies with time in hours according to $T(t) = 1.5t + 10$. Calculate the exact electricity demand after $6$ hours.

A spherical hot-air balloon is inflated using a compressor that injects air at a constant rate, such that its volume in cubic meters 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 time elapsed.

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?

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.