Math Problem Statement

Given that f(x) = 4x + 3 and g(x) = x^2 - 4x - 5, find (g ◦ f)(5).

Solution

To solve (gf)(5)(g \circ f)(5), which is the composition of functions, follow these steps:

Given:

  • f(x)=4x+3f(x) = 4x + 3
  • g(x)=x24x5g(x) = x^2 - 4x - 5

We compute (gf)(5)(g \circ f)(5), which means: (gf)(5)=g(f(5)).(g \circ f)(5) = g(f(5)).

Step 1: Compute f(5)f(5)

f(5)=4(5)+3=20+3=23.f(5) = 4(5) + 3 = 20 + 3 = 23.

Step 2: Compute g(f(5))=g(23)g(f(5)) = g(23)

Substitute x=23x = 23 into g(x)=x24x5g(x) = x^2 - 4x - 5: g(23)=(23)24(23)5.g(23) = (23)^2 - 4(23) - 5.

Step 3: Simplify

g(23)=529925=52997=432.g(23) = 529 - 92 - 5 = 529 - 97 = 432.

Final Answer:

(gf)(5)=432.(g \circ f)(5) = 432.

Would you like me to expand this explanation further or verify any step?

Here are 5 related questions to deepen your understanding:

  1. How is the composition of functions defined in general?
  2. How do you compute (fg)(x)(f \circ g)(x) instead of (gf)(x)(g \circ f)(x)?
  3. What are the domains of f(x)f(x) and g(x)g(x) in this problem?
  4. How do you graphically represent function compositions?
  5. How does function composition differ from regular multiplication of functions?

Tip: Always compute inside-out when solving composition problems!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Function Composition
Linear Functions
Quadratic Functions

Formulas

f(x) = 4x + 3
g(x) = x^2 - 4x - 5
Composition formula: (g ◦ f)(x) = g(f(x))

Theorems

-

Suitable Grade Level

Grades 9-11