Math Problem Statement

If f(x) = x^2 and g(x) = x - 1, find g(f(x)).

Solution

The image contains a problem regarding composition of functions. It states the following:

Given:

  • f(x)=x2f(x) = x^2
  • g(x)=x1g(x) = x - 1

The task is to find g(f(x))g(f(x)).

To solve this:

  1. Start with f(x)=x2f(x) = x^2.
  2. Now, substitute f(x)f(x) into g(x)g(x). This means replacing xx in g(x)=x1g(x) = x - 1 with f(x)=x2f(x) = x^2.

Thus: g(f(x))=g(x2)=x21g(f(x)) = g(x^2) = x^2 - 1

So, the complete expression is: g(f(x))=x21g(f(x)) = x^2 - 1

Would you like further clarification on this? Here are 5 related questions for further practice:

  1. What is the composition of f(g(x))f(g(x)) for the same functions f(x)f(x) and g(x)g(x)?
  2. If f(x)=3x+2f(x) = 3x + 2 and g(x)=2x5g(x) = 2x - 5, what is g(f(x))g(f(x))?
  3. How do you find the inverse of a function, and what is the inverse of g(x)=x1g(x) = x - 1?
  4. What are the domain and range of the composed function g(f(x))g(f(x))?
  5. How do we differentiate the composite function g(f(x))g(f(x))?

Tip: When composing functions, always substitute the entire function inside the variable of the other function, treating it as a replacement for the variable.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Algebra

Formulas

g(f(x)) = g(f(x)) = x^2 - 1

Theorems

Function Composition Theorem

Suitable Grade Level

Grades 9-10