Math Problem Statement

Given that f(x) = 7x + 9 and g(x) = 2 - x^2, calculate (a) f(g(0)) and (b) g(f(0)).

Solution

We are given the functions:

f(x)=7x+9andg(x)=2x2f(x) = 7x + 9 \quad \text{and} \quad g(x) = 2 - x^2

Now, we need to compute the following:

(a) f(g(0))f(g(0))

  1. First, calculate g(0)g(0): g(0)=2(0)2=2g(0) = 2 - (0)^2 = 2
  2. Next, substitute g(0)=2g(0) = 2 into the function f(x)f(x): f(g(0))=f(2)=7(2)+9=14+9=23f(g(0)) = f(2) = 7(2) + 9 = 14 + 9 = 23 So, f(g(0))=23f(g(0)) = 23.

(b) g(f(0))g(f(0))

  1. First, calculate f(0)f(0): f(0)=7(0)+9=9f(0) = 7(0) + 9 = 9
  2. Now, substitute f(0)=9f(0) = 9 into the function g(x)g(x): g(f(0))=g(9)=2(9)2=281=79g(f(0)) = g(9) = 2 - (9)^2 = 2 - 81 = -79 So, g(f(0))=79g(f(0)) = -79.

Final answers:

  • f(g(0))=23f(g(0)) = 23
  • g(f(0))=79g(f(0)) = -79

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

Here are 5 related questions to practice:

  1. Find f(g(1))f(g(1)).
  2. Find g(f(1))g(f(1)).
  3. If h(x)=x2+3xh(x) = x^2 + 3x, what is h(f(2))h(f(2))?
  4. What is the derivative of f(x)=7x+9f(x) = 7x + 9?
  5. Solve for xx if g(x)=0g(x) = 0.

Tip: When working with composite functions like f(g(x))f(g(x)), always evaluate the inner function first and then substitute its result into the outer function.

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Composite Functions

Formulas

f(x) = 7x + 9
g(x) = 2 - x^2

Theorems

Function Composition

Suitable Grade Level

Grades 9-12