Math Problem Statement

Show that f(x) = x^3/2 and g(x) = ³√(2x) are inverse functions algebraically.

Solution

The problem shown in the image is asking to prove algebraically that the functions f(x)=x32f(x) = \frac{x^3}{2} and g(x)=2x3g(x) = \sqrt[3]{2x} are inverse functions. To do this, we need to show that f(g(x))=xf(g(x)) = x and g(f(x))=xg(f(x)) = x.

Let's begin by evaluating f(g(x))f(g(x)) and g(f(x))g(f(x)) step by step.

1. Prove f(g(x))=xf(g(x)) = x:

We are given:

  • f(x)=x32f(x) = \frac{x^3}{2}
  • g(x)=2x3g(x) = \sqrt[3]{2x}

Now, substitute g(x)g(x) into f(x)f(x): [ f(g(x)) = f\left( \sqrt[3]{2x} \right) = \frac{\left( \sqrt[3]{2x} \right)^3}{2} ]

Simplify: f(g(x))=2x2=xf(g(x)) = \frac{2x}{2} = x

Thus, f(g(x))=xf(g(x)) = x, which is correct.

2. Prove g(f(x))=xg(f(x)) = x:

Now, substitute f(x)f(x) into g(x)g(x): g(f(x))=g(x32)=2x323g(f(x)) = g\left( \frac{x^3}{2} \right) = \sqrt[3]{2 \cdot \frac{x^3}{2}}

Simplify: g(f(x))=x33=xg(f(x)) = \sqrt[3]{x^3} = x

Thus, g(f(x))=xg(f(x)) = x, which is also correct.

Conclusion:

Both conditions f(g(x))=xf(g(x)) = x and g(f(x))=xg(f(x)) = x are satisfied, proving that f(x)f(x) and g(x)g(x) are indeed inverse functions.


Do you want more details or explanations on this? Here are 5 related questions to explore:

  1. What is the graphical interpretation of inverse functions?
  2. How do you find the inverse of a function in general?
  3. Can a function have more than one inverse?
  4. What are the conditions for two functions to be inverses?
  5. How can inverse functions be verified numerically?

Tip: Always check both f(g(x))f(g(x)) and g(f(x))g(f(x)) when proving functions are inverses.

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Math Problem Analysis

Mathematical Concepts

Algebra
Inverse Functions
Function Composition

Formulas

f(g(x)) = x
g(f(x)) = x

Theorems

Inverse function property

Suitable Grade Level

Grades 10-12