Math Problem Statement

If f(x)=x 2 −2, g(x)=3x−4, and h(x)=2−4x , find (f−g)(x).

Solution

To find (fg)(x)(f - g)(x), you need to subtract the function g(x)g(x) from f(x)f(x):

Given:

  • f(x)=x22f(x) = x^2 - 2
  • g(x)=3x4g(x) = 3x - 4

Now, (fg)(x)(f - g)(x) is defined as:

(fg)(x)=f(x)g(x)(f - g)(x) = f(x) - g(x)

Substitute the expressions for f(x)f(x) and g(x)g(x):

(fg)(x)=(x22)(3x4)(f - g)(x) = (x^2 - 2) - (3x - 4)

Distribute the negative sign:

(fg)(x)=x223x+4(f - g)(x) = x^2 - 2 - 3x + 4

Simplify the expression:

(fg)(x)=x23x+2(f - g)(x) = x^2 - 3x + 2

So, the result is:

(fg)(x)=x23x+2(f - g)(x) = x^2 - 3x + 2

Let me know if you'd like more details or have any questions! Here are five related questions:

  1. How do you find the difference of two functions in general?
  2. What is the domain of the resulting function (fg)(x)(f - g)(x)?
  3. How would you find the zeros of (fg)(x)(f - g)(x)?
  4. How does subtracting functions relate to adding functions?
  5. How do function transformations affect the graph of (fg)(x)(f - g)(x)?

Tip: Always simplify the resulting expression when combining functions, especially when subtracting or adding!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Function Subtraction
Quadratic Functions
Linear Functions

Formulas

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

Theorems

Function Subtraction Property

Suitable Grade Level

Grades 8-10