Lesson 5 — Function Composition and Inverse Functions

Let $f(x) = 3x + 1$ and $g(x) = x^2$. Compute $(f \circ g)(x)$.

Using the same $f, g$ as in the previous exercise, compute $(g \circ f)(x)$ and compare with $(f \circ g)(x)$.

Let $f(x) = 2x - 5$ and $g(x) = x^2 + 1$. Compute $(f \circ g)(x)$ and $(g \circ f)(x)$.

For $f(x) = x^2 - 12$ and $g(x) = x + 3$, compute $(f \circ g)(2)$.

Let $f(x) = 3x - 2$ and $g(x) = 1/x$. Compute $(f \circ g)(x)$, state the domain, and evaluate $(f \circ g)(2)$ and $(g \circ f)(2)$.

Let $f(x) = 1/(x - 2)$ and $g(x) = x + 3$. Compute $(f \circ g)(x)$ and state the domain.

Let $f(x) = 3x + 2$ and $g(x) = \sqrt{x}$. Compute $(f \circ g)(x)$ and determine the domain.

Let $f(x) = \sqrt{x}$ and $g(x) = x^2 - 4$. Determine $(f \circ g)(x)$ and its domain.

For $f(x) = 1/(x+1)$ and $g(x) = 1/(x+1)$, determine the domain of $(f \circ g)$.

What is the domain of $(f \circ g)(x)$ when $f(x) = \sqrt{x}$ and $g(x) = x - 2$?

Decompose $h(x) = (3x + 2)^4$ as a composition $f \circ g$ of two simpler functions.

Decompose $h(x) = \sqrt{x^2 + 1}$ as a composition $f \circ g$.

Decompose $h(x) = \sqrt{5x + 1}$ as a composition of three functions $h_3 \circ h_2 \circ h_1$.

Decompose $h(x) = \sqrt[3]{1 - x^2}$ as a composition $f \circ g$.

Decompose $h(x) = e^{2x - 5}$ as a composition and determine the domain.

Let $f, g$ be such that $(f \circ g)(x) = x^2 + 4x$ and $g(x) = x + 2$. Determine $f(x)$.

Determine $f(x)$ given that $f(x + 1) = 2x^2 + 3x - 1$.

Which of the following decompositions is correct for $h(x) = 1/(x+3)^2$ as a composition of three functions $h_3 \circ h_2 \circ h_1$?

Find $f^{-1}(x)$ for $f(x) = 3x + 7$.

Find $f^{-1}(x)$ for $f(x) = (x - 1)/2$.

Find $f^{-1}(x)$ for $f(x) = \sqrt[3]{x + 5}$.

Find $f^{-1}(x)$ for $f: [0, +\infty) \to [2, +\infty)$, $f(x) = x^2 + 2$.

Find $f^{-1}(x)$ for $f(x) = (2x + 3)/(x - 1)$, $x \neq 1$.

Verify that $f(x) = 2x + 4$ and $g(x) = (x - 4)/2$ are inverses by computing $f \circ g$ and $g \circ f$.

$f(x) = x^2$ is not invertible on $\mathbb{R}$. Find two different restricted domains where $f$ becomes invertible and exhibit the two inverses.

$f(x) = x^2 - 4x + 7$ is not invertible on $\mathbb{R}$. Restrict the domain to the natural increasing branch and determine $f^{-1}$.

Explain geometrically why the graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y = x$.

How do you decide graphically whether $f$ has an inverse? Describe the criterion and give an example of a function that passes the criterion and one that does not.

Show that $f(x) = a/x$ (with $a \neq 0$, $x \neq 0$) is its own inverse. Functions with this property are called involutions.

Show that $f(x) = 1 - x$ is an involution.

In logistics, the shipping cost is $C(p) = 30 + 4p$ (dollars per kg). Determine $C^{-1}$: what weight corresponds to a shipping cost of $c$ dollars? For a shipping cost of $90, what is the weight?

Celsius to Fahrenheit conversion: $F(C) = (9/5)C + 32$. (a) Determine $F^{-1}$. (b) Compute the temperature in °C corresponding to $F = 100\ °F$.

BRL-to-USD converter: $D(R) = R/5$ (simplified rate). Find $D^{-1}$ and compute how many reais correspond to US$50.

Z-score normalization: $g(x) = x - \bar{x}$ (centering) and $f(y) = y/\sigma$ (scaling). (a) Express the composite $(f \circ g)(x)$. (b) Determine the inverse $(f \circ g)^{-1}$ to back-transform model predictions. Watch the order.

Pharmacokinetics: dose $D$ (mg) produces concentration $C(D) = 0.05\,D$ mg/L. Determine $C^{-1}$: what dose produces concentration $c$? For $c = 2$ mg/L, what is the dose?

A product costs $p$ dollars. Store A: $f(p) = 0.9p$ (10% discount). Store B: $g(p) = p - 50$ ($50 off). (a) For$p = 800$: which is cheaper? (b) For which$p$ do both strategies give the same price?

A pool fills at $V(t) = 80\,t$ liters. Determine $V^{-1}$: how long to fill $v$ liters? For 4,000 L?

Chain conversion: USD → BRL via $f(d) = 5d$ (simplified rate); BRL → BTC via $g(r) = r/350\,000$. (a) Model USD → BTC as the composite $g \circ f$. (b) Determine the inverse BTC → USD. (c) How many dollars is 0.01 BTC worth?

Prove that if $f$ is bijective, then $(f^{-1})^{-1} = f$.

Prove that the composition of two injective functions is injective.

Prove that if $f$ and $g$ are bijective, then $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$.

If $f \circ g$ is injective, prove that $g$ is injective. Is the converse true for $f$? Justify with a counterexample.

Caesar cipher. Encoding: $E_k(\ell) = (\ell + k) \bmod 26$ for $\ell \in \{0, \ldots, 25\}$ and shift $k$. (a) Determine $E_k^{-1}$. (b) For $k = 3$, encode "H" (= 7) and verify that decryption recovers "H".

For bijective $f$, determine $((f^{-1})^{-1})^{-1}$. Justify using the uniqueness of the inverse.

An involution is a function $f$ with $f \circ f = \operatorname{id}$. Show that involutions are self-inverse and verify that $f(x) = c - x$, $f(x) = -x$, and $f(x) = 1/x$ are examples.