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Chapter 4: Problem 37

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

Short Answer

Expert verified
(f-h)(-3) = 6

Step by step solution

01

Understand the problem

We need to find \( (f-h)(-3) \). This means finding the difference between the functions \(f(x)\) and \(h(x)\) at \ x = -3 \.
02

Find \(f(-3)\)

First, we need to evaluate \(f(x) = x^2 - 9\) at \(x = -3\). So, \(f(-3) = (-3)^2 - 9 = 9 - 9 = 0\).
03

Find \(h(-3)\)

Next, we need to evaluate \(h(x) = x - 3\) at \(x = -3\). So, \(h(-3) = -3 - 3 = -6\).
04

Calculate \( (f-h)(-3) \)

Finally, subtract the values from the previous steps: \( (f-h)(-3) = f(-3) - h(-3) = 0 - (-6) = 6 \).

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Key Concepts

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

Polynomial Functions
Polynomial functions are expressions involving variables and coefficients, structured as the sum of terms where each term is the product of a constant and a variable raised to a non-negative integer power.
For example, the functions given in the exercise, such as \(f(x) = x^2 - 9\), exemplify polynomial functions.
Here are a few key points about polynomial functions:
  • They are smooth and continuous curves.
  • The degree of a polynomial is determined by the highest exponent of the variable.
  • Polynomial functions can be simple linear (degree 1) or more complex quadratic (degree 2), cubic (degree 3), etc.
Understanding polynomial functions is crucial when it comes to solving problems that involve them, like the one in the exercise.
Evaluating Functions
Evaluating functions means finding the output value given an input value. To evaluate a function, you substitute the input value into the function and simplify.
Using the exercise example, we had to evaluate both \(f(x)\) and \(h(x)\) at \(x = -3\).
Here is the step-by-step process used:
  • First, evaluate \(f(x) = x^2 - 9\). When \(x = -3\), \(f(-3) = (-3)^2 - 9 = 9 - 9 = 0\).
  • Next, evaluate \(h(x) = x - 3\). When \(x = -3\), \(h(-3) = -3 - 3 = -6\).
Evaluating functions properly ensures that you find the correct values to use for further calculations, like function subtraction.
Function Subtraction
Function subtraction involves subtracting one function from another and then evaluating the result at a given point. Given two functions \(f(x)\) and \(h(x)\), the subtraction of these functions is expressed as \((f-h)(x)\).
In the exercise, we are asked to find \((f-h)(-3)\). Here are the steps followed:
  • First, find \(f(x)\) and \(h(x)\) values at \(x = -3\), resulting in 0 and -6, respectively.
  • Then, subtract these values: \((f-h)(-3) = f(-3) - h(-3) = 0 - (-6) = 6\).
It鈥檚 important to handle the subtraction carefully, especially with negative values, to avoid mistakes. Subtraction of functions allows us to see the difference in their values at specific points.

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Most popular questions from this chapter

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \frac{\left(m^{4}\right)^{-1}}{9 m^{3}} $$

Find each product. $$ (m-5 p)\left(m^{2}-2 m p+3 p^{2}\right) $$

Find each product. $$ [((3 m-y)+z][(3 m-y)-z] $$

The factors in the following exercises involve fractions or decimals. Apply the methods of this section, and find each product. $$ \left(4 x-\frac{2}{3}\right)\left(4 x+\frac{2}{3}\right) $$

Find each product. $$ (4 n-3 m)^{2} $$
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