Problem 23 - Algebra Form and Function

Question

Check that the functions are inverses. $$ f(x)=32 x^{5}-2 \text { and } g(t)=\frac{(t+2)^{1 / 5}}{2} $$

Step-by-Step Solution

Answer

Answer: Yes, the functions are inverses of each other since $$f(g(t)) = t$$ and $$g(f(x)) = x$$.

Step 1: Find the composition f(g(t))

To find f(g(t)), substitute g(t) into f(x): $$ f(g(t)) = 32\left(\frac{(t+2)^{1/5}}{2}\right)^5 - 2 $$

Step 2: Simplify f(g(t))

Let's simplify the equation by applying the power to the g(t) function: $$ f(g(t)) = 32\left(\frac{(t+2)^{1}}{32}\right) - 2 $$ Cancel out the 32 with the denominator, $$ f(g(t)) = t + 2 - 2 $$ Simplify further, $$ f(g(t)) = t $$

Step 3: Find the composition g(f(x))

To find g(f(x)), substitute f(x) into g(t): $$ g(f(x)) = \frac{(32 x^5 - 2 + 2)^{1/5}}{2} $$

Step 4: Simplify g(f(x))

Let's simplify the equation by removing the constants from the parenthesis: $$ g(f(x)) = \frac{(32 x^5)^{1/5}}{2} $$ Use the property of exponentiation to simplify this further, $$ g(f(x)) = \frac{2x}{2} $$ The denominator cancels with the numerator, $$ g(f(x)) = x $$ Since f(g(t)) = t and g(f(x)) = x, we have proven that f(x) and g(t) are inverse functions.

Key Concepts

Function Composition

Function composition is like building a machine inside another. To check if two functions are inverses, we often perform function composition. This means you plug the output of one function into another function. In math, this is represented as \( f(g(t)) \) and \( g(f(x)) \). If after plugging one function into the other, you end up back where you started, then the functions are inverses.

For instance, to determine if \( f(x) = 32x^5 - 2 \) and \( g(t) = \frac{(t+2)^{1/5}}{2} \) are inverses, you compute \( f(g(t)) \) and \( g(f(x)) \).

• In \( f(g(t)) \), substitute \( g(t) \) where \( x \) appears in \( f(x) \).
• In \( g(f(x)) \), substitute \( f(x) \) where \( t \) appears in \( g(t) \).

If both results simplify to \( t \) and \( x \), respectively, you confirm that they "undo" each other, indicating they are true inverses.

Exponents

Exponents can often seem tricky, but they simplify many mathematical expressions. In this situation, we deal with exponents of 5 and \( \frac{1}{5} \). These represent raising a number to the fifth power and finding the fifth root.Opposite operations like these cancel each other out, another reason they show two functions are inverses.

In our problem, observe:

• \( 32x^5 \) indicates \( x \) multiplied by itself five times, then scaled up by 32.
• \( (t+2)^{1/5} \) calculates the fifth root. When these operations occur together, as in our function pair, they counterbalance, simplifying to return the original number.

On simplification of exponent operations in the compositions, the base emerges untouched, helping confirm the inverse relationships.

Simplification

Simplification means breaking down complicated expressions into simpler forms. This involves arithmetic operations - adding, subtracting, multiplying, dividing - and other actions, like canceling out terms. In mathematics, simpler expressions often reveal underlying truths more clearly.

Through simplification:

• In \( f(g(t)) = 32\left(\frac{(t+2)^{1/5}}{2}\right)^5 - 2 \), operations cancel each other, leaving \( t \), so \( f(g(t)) = t \).
• Similarly, \( g(f(x)) = \frac{(32x^5 - 2 + 2)^{1/5}}{2} \) simplifies to \( x \).

Canceling and reducing helps verify that these functions perfectly invert each other, confirming their inverse nature.

Algebraic Functions

Algebraic functions encompass a wide range of mathematical expressions including polynomials, roots, and rational expressions. The main goal is evaluating how these expressions behave with manipulations such as addition or composition.

Take the algebraic functions we’re dealing with: v

• \( f(x) = 32x^5 - 2 \) is a polynomial function, where power and coefficients dictate its form.
• \( g(t) = \frac{(t+2)^{1/5}}{2} \) employs roots and division, showcasing a different algebraic structure.

When explored through function compositions, these reveal intrinsic algebraic connecting properties. Simplifying shows how algebraic operations combine, and whether these functions invert each other's effect, which completes our analysis of whether they are truly inverses.