Math Problem Statement

For f(x) = x - 3 and g(x) = 5x^2 - 3, find the following functions: a) f ∘ g(x).

Solution

The image contains a problem where two functions are given:

  • f(x)=x3f(x) = x - 3
  • g(x)=5x23g(x) = 5x^2 - 3

We are asked to find the composition of functions fg(x)f \circ g(x), which means we need to evaluate f(g(x))f(g(x)).

Solution for part (a): fg(x)f \circ g(x)

  1. Start with the function for g(x)g(x): g(x)=5x23g(x) = 5x^2 - 3
  2. Now substitute g(x)g(x) into the function f(x)f(x), i.e., replace xx in f(x)=x3f(x) = x - 3 with g(x)g(x): f(g(x))=g(x)3=(5x23)3f(g(x)) = g(x) - 3 = (5x^2 - 3) - 3
  3. Simplify: f(g(x))=5x26f(g(x)) = 5x^2 - 6

Thus, the solution for part (a) is: fg(x)=5x26f \circ g(x) = 5x^2 - 6

Let me know if you would like help with other parts of the problem or further explanations.

Here are 5 related questions you can explore:

  1. How do you calculate gf(x)g \circ f(x)?
  2. What is the composition of functions at a specific value, like f(g(1))f(g(-1))?
  3. Can you explain the difference between fg(x)f \circ g(x) and gf(x)g \circ f(x)?
  4. How do you check if a composition of functions is invertible?
  5. What happens if you switch the order of composition between two arbitrary functions?

Tip: When composing functions, always substitute the entire function for the variable in the outer function!

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Math Problem Analysis

Mathematical Concepts

Algebra
Function Composition
Quadratic Equations

Formulas

f(g(x)) = f(g(x)) = f(g(x)) = x - 3

Theorems

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Suitable Grade Level

Grades 9-11