Math Problem Statement

Determine g(x+a)−g(x) for the following function.

g(x)=4x2−4

Solution

To determine g(x+a)g(x)g(x + a) - g(x) for the given function g(x)=4x24g(x) = 4x^2 - 4, we need to follow these steps:

  1. Compute g(x+a)g(x + a): Substitute x+ax + a into the function:

g(x+a)=4(x+a)24g(x + a) = 4(x + a)^2 - 4

Expand the squared term:

(x+a)2=x2+2ax+a2(x + a)^2 = x^2 + 2ax + a^2

So,

g(x+a)=4(x2+2ax+a2)4=4x2+8ax+4a24g(x + a) = 4(x^2 + 2ax + a^2) - 4 = 4x^2 + 8ax + 4a^2 - 4

  1. Subtract g(x)g(x) from g(x+a)g(x + a):

g(x+a)g(x)=(4x2+8ax+4a24)(4x24)g(x + a) - g(x) = (4x^2 + 8ax + 4a^2 - 4) - (4x^2 - 4)

Simplify by canceling like terms:

g(x+a)g(x)=8ax+4a2g(x + a) - g(x) = 8ax + 4a^2

Final Answer:

g(x+a)g(x)=8ax+4a2g(x + a) - g(x) = 8ax + 4a^2

Would you like further details on any step, or have any questions?

Related Questions:

  1. How would you solve for g(x+2a)g(x)g(x + 2a) - g(x) given the same function?
  2. What is the derivative of g(x)=4x24g(x) = 4x^2 - 4?
  3. How can you interpret the result 8ax+4a28ax + 4a^2 in terms of algebraic functions?
  4. How would the expression change if g(x)=4x24x+1g(x) = 4x^2 - 4x + 1?
  5. What is the second derivative of g(x)=4x24g(x) = 4x^2 - 4?

Tip:

Always double-check the expansion of squared binomials to avoid common algebraic mistakes!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Function Differences

Formulas

g(x) = 4x^2 - 4
(x + a)^2 = x^2 + 2ax + a^2

Theorems

Binomial Expansion

Suitable Grade Level

Grades 9-12