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Chapter 2: Problem 30

Let \(F(x)=x^{2}-2\) and \(G(x)=5-x .\) Find each of the following. $$(F / G)(-1)$$

Short Answer

Expert verified
(F/G)(-1) = -1/6

Step by step solution

01

- Evaluate F(-1)

Start by finding the value of the function F at x = -1. Use the formula for F(x): \[ F(x) = x^2 - 2 \] Substitute -1 for x: \[ F(-1) = (-1)^2 - 2 \]Simplify the expression: \[ F(-1) = 1 - 2 = -1 \]
02

- Evaluate G(-1)

Next, find the value of the function G at x = -1. Use the formula for G(x): \[ G(x) = 5 - x \] Substitute -1 for x: \[ G(-1) = 5 - (-1) \]Simplify the expression: \[ G(-1) = 5 + 1 = 6 \]
03

- Divide F(-1) by G(-1)

Now divide the value of F(-1) by the value of G(-1): \[ \frac{F(-1)}{G(-1)} = \frac{-1}{6} \] Therefore, \[ (F/G)(-1) = \frac{-1}{6} \]

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Key Concepts

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

Evaluating Functions
Functions are like machines. You input a number, and the function processes it to give you an output. This process is called 'evaluating a function'.
To evaluate a function, you need to know the function's formula. Let鈥檚 say we have two functions:
  • \[ F(x) = x^2 - 2 \]
  • \[ G(x) = 5 - x \]
To evaluate these, let's plug in x = -1 and see what we get. For \[ F(x) \], we substitute \[ -1 \] into \[ F \] and simplify:
\[ F(-1) = (-1)^2 - 2 = 1 - 2 = -1 \].
Next, let's do the same for \[ G(x) \]:
\[ G(-1) = 5 - (-1) = 5 + 1 = 6 \].
So, \[ F(-1) = -1 \] and \[ G(-1) = 6 \]. Easy to follow, right? Now let's move on to the next concept.
Function Division
Division of functions might sound a bit tricky, but it鈥檚 just basic algebra. When we talk about dividing functions, it means we take the outputs of each function and divide them.
Using our evaluated functions, we have:\( F(-1) \) and \( G(-1) \) from the previous section:
\[ \frac{F(-1)}{G(-1)} = \frac{-1}{6} \].
This tells us that when we divide the output of \( F \) at \( x = -1 \) by the output of \( G \) at \( x = -1 \), we get \( \frac{-1}{6} \).
Function division isn't always straightforward, but with careful evaluation, it becomes manageable.
Algebraic Simplification
Simplifying algebraic expressions is fundamental in solving function problems. Once you substitute an input into a function and get the output, you often need to simplify the result.
Look at \[ F(x) = x^2 - 2 \] when \[ x = -1 \]:
First, square \[ -1 \]: \[ (-1)^2 = 1 \]. Then, subtract 2: \[ 1 - 2 = -1 \].
This steps how we simplify to get \[ F(-1) = -1 \]. The same process happens for \[ G(x) = 5 - x \]:
Substitute \[ -1 \]: \[ G(-1) = 5 - (-1) \]. Simplify the double negative: \[ 5 + 1 = 6 \]. Simple steps like these make working with functions easier.
Substitution Method
The substitution method is key when dealing with functions. It involves putting a specific value in place of the variable (\[ x \]) and performing the necessary operations.
To illustrate, using our functions \[ F(x) = x^2 - 2 \] and \[ G(x) = 5 - x \], we substitute \[ x = -1 \]:
  • For \[ F(x) \]: Replace \[ x \] with \[ -1 \]: \[ F(-1) = (-1)^2 - 2 = -1 \].
  • For \[ G(x) \]: Replace \[ x \] with \[ -1 \]: \[ G(-1) = 5 - (-1) = 6 \].
The values \[ -1 \] and \[ 6 \] are the results of this substitution. This method makes understanding functions more tangible by focusing on specific values.

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