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Magazine Chapter 2: Problem 2
Find (a) \((f+g)(3)\) (b) \((f-g)(3)\) (c) \((f g)(3)\) (d) \((f / g)(3)\) $$f(x)=-x^{2}, \quad g(x)=2 x-1$$
Short Answer
Expert verified
(a) -4, (b) -14, (c) -45, (d) -1.8
Step by step solution
01
Evaluate f(3)
First, we need to evaluate the function \( f(x) = -x^2 \) at \( x = 3 \). Substitute \( x = 3 \) into \( f(x) \): \[ f(3) = -(3)^2 = -9 \]
02
Evaluate g(3)
Next, we need to evaluate the function \( g(x) = 2x - 1 \) at \( x = 3 \). Substitute \( x = 3 \) into \( g(x) \):\[ g(3) = 2(3) - 1 = 6 - 1 = 5 \]
03
Calculate (f+g)(3)
Using the results from Step 1 and Step 2, calculate \((f+g)(3)\):\[ (f+g)(3) = f(3) + g(3) = -9 + 5 = -4 \]
04
Calculate (f-g)(3)
Now calculate \((f-g)(3)\) using the values from Step 1 and Step 2:\[ (f-g)(3) = f(3) - g(3) = -9 - 5 = -14 \]
05
Calculate (f\cdot g)(3)
Calculate \((f \cdot g)(3)\) by multiplying the results from Step 1 and Step 2:\[ (f \cdot g)(3) = f(3) \cdot g(3) = (-9) \cdot 5 = -45 \]
06
Calculate (f/g)(3)
Finally, calculate \((f/g)(3)\) by dividing the result from Step 1 by the result from Step 2:\[ \left( \frac{f}{g} \right)(3) = \frac{f(3)}{g(3)} = \frac{-9}{5} = -1.8 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Evaluating Functions
To evaluate a function means to find the output for a given input. Let's break it down simply. If you have a function such as \( f(x) = -x^2 \), and you want to find \( f(3) \), you substitute \( x \) with 3. So, \( f(3) = -(3)^2 = -9 \).
This is like finding out what happens when you put 3 into the "recipe" that is your function. Similarly, for another function \( g(x) = 2x - 1 \), evaluating \( g(3) \) involves plugging in the value of 3, which gives \( g(3) = 2(3) - 1 = 5 \).
So, whenever you evaluate a function, just replace the variable with the number you have, and calculate the result.
This is like finding out what happens when you put 3 into the "recipe" that is your function. Similarly, for another function \( g(x) = 2x - 1 \), evaluating \( g(3) \) involves plugging in the value of 3, which gives \( g(3) = 2(3) - 1 = 5 \).
So, whenever you evaluate a function, just replace the variable with the number you have, and calculate the result.
Addition of Functions
When you add two functions, you create a new, combined function. Imagine the addition of \( f(x) = -x^2 \) and \( g(x) = 2x - 1 \). The operation \( (f+g)(x) \) is done by simply adding these two functions together.
For our specific case for \( x = 3 \), we calculated:
For our specific case for \( x = 3 \), we calculated:
- \( f(3) = -9 \)
- \( g(3) = 5 \)
Subtraction of Functions
Subtracting functions works just like addition, but instead you subtract the outputs. For example, with \( f(x) = -x^2 \) and \( g(x) = 2x - 1 \), the operation \( (f-g)(x) \) involves subtracting \( g(x) \) from \( f(x) \).
For \( x = 3 \), we just subtract the result of \( g(3) \) from \( f(3) \):
For \( x = 3 \), we just subtract the result of \( g(3) \) from \( f(3) \):
- \( f(3) = -9 \)
- \( g(3) = 5 \)
Multiplication of Functions
Multiplication of functions involves multiplying the outputs of the functions when they share the same input. Sounds simple, right? For our functions \( f(x) = -x^2 \) and \( g(x) = 2x - 1 \), the multiplication operation gives \( (f \cdot g)(x) \).
For \( x = 3 \), this involves:
For \( x = 3 \), this involves:
- Multiplying \( f(3) = -9 \)
- By \( g(3) = 5 \)
Division of Functions
Division of functions may seem tricky at first, but it's really about dividing the outputs of two functions at the same input. Here, for functions \( f(x) = -x^2 \) and \( g(x) = 2x - 1 \), we calculate \( \left(\frac{f}{g}\right)(x) \).
For \( x = 3 \):
For \( x = 3 \):
- Divide \( f(3) = -9 \)
- By \( g(3) = 5 \)
