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Chapter 9: Problem 49

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ (g-h)(-3) $$

Short Answer

Expert verified
(g-h)(-3) = 0

Step by step solution

01

- Find g(-3)

Evaluate the function g at x = -3. Given that g(x) = 2x, substitute -3 for x: \[ g(-3) = 2(-3) = -6 \]
02

- Find h(-3)

Evaluate the function h at x = -3. Given that h(x) = x - 3, substitute -3 for x: \[ h(-3) = -3 - 3 = -6 \]
03

- Calculate (g-h)(-3)

Compute the difference (g - h) at x = -3 using the values obtained for g(-3) and h(-3) from the previous steps: \[ (g-h)(-3) = g(-3) - h(-3) = -6 - (-6) \] Simplify the expression: \[ (g-h)(-3) = -6 + 6 = 0 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Evaluation
Let's start by understanding how to evaluate functions. Function evaluation means finding the value of a function at a specific point.
For example, if you have a function g(x) = 2x and you want to find g(-3), you simply substitute -3 for x in the function.
So, g(-3) = 2(-3) = -6.
This tells you that when x is -3, the value of the function g is -6.
Function evaluation is a fundamental concept in algebra and helps in performing operations on functions like addition, subtraction, multiplication, and division.
Difference of Functions
Next, let's explore the difference of functions. The difference of two functions (f - g)(x) means subtracting one function from another.
For example, if you have two functions g(x) = 2x and h(x) = x - 3, and you want to find (g - h)(-3), you first evaluate each function at x = -3.
From the given functions:
  • g(-3) = 2(-3) = -6
  • h(-3) = -3 - 3 = -6
Now, subtract the value of h(-3) from g(-3): g(-3) - h(-3) = -6 - (-6).
Simplifying this, you get -6 + 6 = 0. So, (g - h)(-3) = 0.
This step is about managing the signs correctly and performing the algebraic subtraction accurately.
Simplification Steps
Finally, let's focus on the simplification steps. Simplification in function operations often involves combining like terms and correctly handling positive and negative signs.
Using our example, after evaluating g(-3) and h(-3), we got: g(-3) = -6 and h(-3) = -6.
When we subtract h(-3) from g(-3), we have: -6 - (-6)
It's crucial to remember that subtracting a negative number is the same as adding its positive counterpart. So, -6 - (-6) turns into -6 + 6.
This simplifies to 0: -6 + 6 = 0.
Simplification ensures that your final answer is in its simplest form. Breaking down steps carefully helps avoid mistakes and ensures that the solution is accurate.

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Most popular questions from this chapter

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