Problem 4

Question

Show that $f$ and $g$ are inverse functions (a) analytically and (b) graphically. $f(x)=1-x^{3}, \quad g(x)=\sqrt[3]{1-x}$

Step-by-Step Solution

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Answer

Analytically, both $g(f(x))$ and $f(g(x))$ simplify to $x$, satisfying the property of inverse functions. Graphically, $f$ and $g$ are reflections of each other over the line $y=x$, so they are indeed inverse functions.

1Step 1: Analytical proof

Substitute $f(x)$ into $g(x)$ to yield $g(f(x))$. Substituting $1 - x^3$ into $g$ gives us $g(f(x)) = \sqrt[3]{1 - (1 - x^3)}$, which simplifies to $g(f(x)) = \sqrt[3]{x^3}$, or $x$. Repeat the process by substituting $g(x)$ into $f(x)$ to get $f(g(x)) = 1 - (\sqrt[3]{1 - x})^3$, which also simplifies to $x$.

2Step 2: Graphical proof

First, plot the function $f(x) = 1 - x^3$. This will be a cubic function decreasing from $f(x) = 1$ when $x = 0$ to $f(x) = - \infty$ as $x$ approaches $ \infty$. Secondly, plot $g(x) = \sqrt[3]{1 - x}$, looking like the left part of the cubic function starting at $x = 1$. As a final step, draw the line $y = x$. The graph of $g$ should be the reflection of $f$ over this line, proving that they are indeed inverse functions.

Key Concepts

Analytical Proof of Inverse FunctionsGraphical Proof of Inverse FunctionsComposition of FunctionsReflection Over Line $ y=x $

Analytical Proof of Inverse Functions

To show that two functions are inverses of each other analytically, we need to perform a composition of functions. This means we substitute one function into the other and simplify it. If both compositions simplify to the variable itself, usually noted as 'x', then the functions are inverses.

For example, consider functions $ f(x) = 1 - x^3 $ and $ g(x) = \sqrt[3]{1-x} $. First, we check $ g(f(x)) $. Substituting $ f(x) $ into $ g(x) $ gives:

$ g(f(x)) = \sqrt[3]{1 - (1 - x^3)} $
Simplifying, we have $ g(f(x)) = \sqrt[3]{x^3} = x $

Next, we check $ f(g(x)) $. Substituting $ g(x) $ into $ f(x) $ gives:

$ f(g(x)) = 1 - (\sqrt[3]{1-x})^3 $
And simplifying, we have $ f(g(x)) = x $

Both compositions simplify to 'x', confirming that $ f $ and $ g $ are inverse functions.

Graphical Proof of Inverse Functions

Graphical proof provides a visual way of confirming that two functions are inverses. This involves plotting both functions and observing their relationship to the line $ y = x $.

For the functions $ f(x) = 1 - x^3 $ and $ g(x) = \sqrt[3]{1-x} $, we begin by plotting $ f(x) $, which is a cubic function that decreases with increasing $ x $. Next, we plot $ g(x) $, which resembles a partial cubic curve beginning at $ x = 1 $. When both graphs are drawn, along with the line $ y = x $, we notice something important:

If $ g(x) $ is a mirror image of $ f(x) $ over the line $ y = x $, then they are inverse functions.

This reflection confirms their inverse relationship. A graphical approach helps in understanding the symmetrical nature of inverse functions.

Composition of Functions

The composition of functions involves taking the output of one function and using it as the input for another. Composing functions is a crucial step in proving whether they are inverses of each other.

For instance, if we have two functions, $ f(x) $ and $ g(x) $, computing the compositions $ g(f(x)) $ and $ f(g(x)) $ is essential. If both results equal 'x', it confirms the inverse nature.

In our case:

$ g(f(x)) $ simplifies to 'x'
$ f(g(x)) $ also simplifies to 'x'

This symmetric return to 'x' means the input is restored, proving the functions’ inverse nature. Consider composition of functions as an algebraic method to verify inverses.

Reflection Over Line $ y=x $

The concept of reflection over the line $ y = x $ is a graphical way to see inverses in action. When a function and its inverse are graphed, each should reflect over this diagonal line, acting as a mirror that confirms their relationship.

Visualize $ y = x $ as a line of symmetry. If a point $ (a, b) $ on the graph of function $ f $ can reflect over this line to become $ (b, a) $ on the graph of $ g $, then they are inverses: