Given That $f(x) = X^2 - 17$ And $g(x) = 5 - X$, Find $ ( F − G ) ( 3 ) (f-g)(3) ( F − G ) ( 3 ) [/tex], If It Exists.Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. $(f-g)(3) =$
When dealing with functions, we often need to perform operations such as addition, subtraction, multiplication, and division. In this problem, we are given two functions, f(x) and g(x), and we are asked to find the value of (f-g)(3), which is the result of subtracting g(x) from f(x) and evaluating the resulting function at x=3.
Defining the Functions
The two functions given are:
- f(x) = x^2 - 17
- g(x) = 5 - x
To find (f-g)(3), we need to first find the value of f(3) and g(3).
Evaluating f(3)
To evaluate f(3), we substitute x=3 into the function f(x):
f(3) = (3)^2 - 17 f(3) = 9 - 17 f(3) = -8
Evaluating g(3)
To evaluate g(3), we substitute x=3 into the function g(x):
g(3) = 5 - 3 g(3) = 2
Finding (f-g)(3)
Now that we have found the values of f(3) and g(3), we can find the value of (f-g)(3) by subtracting g(3) from f(3):
(f-g)(3) = f(3) - g(3) (f-g)(3) = -8 - 2 (f-g)(3) = -10
Therefore, the value of (f-g)(3) is -10.
Conclusion
In this problem, we were given two functions, f(x) and g(x), and we were asked to find the value of (f-g)(3). We first evaluated f(3) and g(3), and then found the value of (f-g)(3) by subtracting g(3) from f(3). The result was -10.
Key Takeaways
- When subtracting functions, we need to evaluate the functions at the given value of x.
- The result of subtracting functions is a new function, which we can evaluate at different values of x.
- In this problem, we used the values of f(3) and g(3) to find the value of (f-g)(3).
Practice Problems
- Find the value of (f+g)(3) if f(x) = x^2 - 17 and g(x) = 5 - x.
- Find the value of (f-g)(2) if f(x) = 2x^2 - 3 and g(x) = x - 1.
Solutions
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To find the value of (f+g)(3), we need to first find the values of f(3) and g(3). Then, we can add the values of f(3) and g(3) to find the value of (f+g)(3).
f(3) = (3)^2 - 17 f(3) = 9 - 17 f(3) = -8
g(3) = 5 - 3 g(3) = 2
(f+g)(3) = f(3) + g(3) (f+g)(3) = -8 + 2 (f+g)(3) = -6
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To find the value of (f-g)(2), we need to first find the values of f(2) and g(2). Then, we can subtract the value of g(2) from the value of f(2) to find the value of (f-g)(2).
f(2) = 2(2)^2 - 3 f(2) = 2(4) - 3 f(2) = 8 - 3 f(2) = 5
g(2) = 2 - 1 g(2) = 1
(f-g)(2) = f(2) - g(2) (f-g)(2) = 5 - 1 (f-g)(2) = 4
Q&A: Subtracting Functions =============================
In this article, we will answer some common questions related to subtracting functions.
Q: What is the difference between subtracting functions and subtracting numbers?
A: When we subtract numbers, we are finding the difference between two numbers. For example, 5 - 3 = 2. However, when we subtract functions, we are finding the difference between two functions. For example, (f-g)(x) = f(x) - g(x). This means that we need to evaluate the functions at the given value of x and then subtract the values.
Q: How do I evaluate a function at a given value of x?
A: To evaluate a function at a given value of x, we need to substitute the value of x into the function. For example, if we have the function f(x) = x^2 - 17 and we want to evaluate it at x=3, we would substitute x=3 into the function:
f(3) = (3)^2 - 17 f(3) = 9 - 17 f(3) = -8
Q: What is the result of subtracting two functions?
A: The result of subtracting two functions is a new function, which we can evaluate at different values of x. For example, if we have the functions f(x) = x^2 - 17 and g(x) = 5 - x, the result of subtracting g(x) from f(x) is:
(f-g)(x) = f(x) - g(x) (f-g)(x) = x^2 - 17 - (5 - x) (f-g)(x) = x^2 - 17 - 5 + x (f-g)(x) = x^2 + x - 22
Q: Can I subtract functions with different variables?
A: Yes, you can subtract functions with different variables. For example, if we have the functions f(x) = x^2 - 17 and g(y) = 5 - y, we can subtract g(y) from f(x):
(f-g)(x) = f(x) - g(y) (f-g)(x) = x^2 - 17 - (5 - y) (f-g)(x) = x^2 - 17 - 5 + y (f-g)(x) = x^2 + y - 22
However, we need to be careful when subtracting functions with different variables. We need to make sure that the variables are the same before we can subtract the functions.
Q: Can I subtract functions with different domains?
A: No, you cannot subtract functions with different domains. The domain of a function is the set of all possible input values for the function. If two functions have different domains, it means that they are not defined for the same values of x. Therefore, we cannot subtract the functions.
For example, if we have the functions f(x) = x^2 - 17 and g(x) = 1/x, we cannot subtract g(x) from f(x) because the domain of g(x) is all real numbers except 0, while the domain of f(x) is all real numbers.
Q: Can I subtract functions with different ranges?
A: No, you cannot subtract functions with different ranges. The range of a function is the set of all possible output values for the function. If two functions have different ranges, it means that they produce different values for the same input. Therefore, we cannot subtract the functions.
For example, if we have the functions f(x) = x^2 - 17 and g(x) = 5 - x, we cannot subtract g(x) from f(x) because the range of f(x) is all real numbers, while the range of g(x) is all real numbers except 5.
Conclusion
In this article, we have answered some common questions related to subtracting functions. We have discussed how to evaluate a function at a given value of x, what the result of subtracting two functions is, and whether we can subtract functions with different variables, domains, or ranges. We hope that this article has been helpful in understanding the concept of subtracting functions.