Math Problem Statement

Composite f(g(x)) g=x-5 f=x^2+3

Solution

To find the composite function f(g(x))f(g(x)), we substitute g(x)g(x) into f(x)f(x).

Given:

  • g(x)=x5g(x) = x - 5
  • f(x)=x2+3f(x) = x^2 + 3

Now, for the composite function f(g(x))f(g(x)), substitute g(x)g(x) into f(x)f(x): f(g(x))=f(x5)f(g(x)) = f(x - 5)

Since f(x)=x2+3f(x) = x^2 + 3, replace xx in f(x)f(x) with x5x - 5: f(g(x))=(x5)2+3f(g(x)) = (x - 5)^2 + 3

Now, expand (x5)2(x - 5)^2: (x5)2=x210x+25(x - 5)^2 = x^2 - 10x + 25

Thus: f(g(x))=x210x+25+3=x210x+28f(g(x)) = x^2 - 10x + 25 + 3 = x^2 - 10x + 28

So, the composite function is: f(g(x))=x210x+28f(g(x)) = x^2 - 10x + 28

Would you like more details or have any questions about this process?

Here are 5 related questions to expand this topic:

  1. How do you find the composite function if f(x)f(x) and g(x)g(x) are different?
  2. What is the domain of the composite function f(g(x))f(g(x))?
  3. How would the composite function change if f(x)=x23f(x) = x^2 - 3?
  4. What is the difference between f(g(x))f(g(x)) and g(f(x))g(f(x))?
  5. How does the order of composition affect the resulting function?

Tip: Always check if the functions are defined for all xx in the domain before composing them.

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Math Problem Analysis

Mathematical Concepts

Algebra
Composite Functions
Quadratic Equations

Formulas

f(g(x)) = (x - 5)^2 + 3
(x - 5)^2 = x^2 - 10x + 25

Theorems

Composite Function Theorem
Quadratic Expansion

Suitable Grade Level

Grades 9-12