Math Problem Statement

Find (f o g)(4) when f(x) = -3x - 4 and g(x) = 2x^2 - 2x + 7.

Solution

The question asks to find (fg)(4)(f \circ g)(4), where:

  • f(x)=3x4f(x) = -3x - 4
  • g(x)=2x22x+7g(x) = 2x^2 - 2x + 7

We are asked to first calculate g(4)g(4) and then plug the result into f(x)f(x).

Step 1: Calculate g(4)g(4)

Using the function g(x)=2x22x+7g(x) = 2x^2 - 2x + 7:

g(4)=2(4)22(4)+7g(4) = 2(4)^2 - 2(4) + 7 g(4)=2(16)8+7=328+7=31g(4) = 2(16) - 8 + 7 = 32 - 8 + 7 = 31

Step 2: Calculate f(g(4))=f(31)f(g(4)) = f(31)

Now, using f(x)=3x4f(x) = -3x - 4, we plug in g(4)=31g(4) = 31:

f(31)=3(31)4f(31) = -3(31) - 4 f(31)=934=97f(31) = -93 - 4 = -97

Thus, (fg)(4)=97(f \circ g)(4) = -97.

The correct answer is D. -97.

Would you like more details on any part of this solution?

Related Questions:

  1. What is the domain of fg(x)f \circ g(x)?
  2. Can you find (gf)(4)(g \circ f)(4) using the same functions?
  3. How do the graphs of f(x)f(x) and g(x)g(x) differ?
  4. What is the range of f(x)f(x) given the domain x0x \geq 0?
  5. How would the composition (fg)(x)(f \circ g)(x) change if g(x)g(x) was a linear function?

Tip:

When solving function compositions, always start by evaluating the innermost function first before substituting its result into the outer function.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Quadratic Functions
Linear Functions

Formulas

g(x) = 2x^2 - 2x + 7
f(x) = -3x - 4

Theorems

-

Suitable Grade Level

Grades 9-12