For The Function $f(x) = 4(x+3$\], Find $f^{-1}(x$\].A. $f^{-1}(x) = 4x - 3$ B. $f^{-1}(x) = \frac{(x+3)}{4}$ C. $f^{-1}(x) = \frac{x}{4} - 3$ D. $f^{-1}(x) = \frac{(x-3)}{4}$
Introduction
In mathematics, an inverse function is a function that reverses the operation of another function. In other words, if we have a function f(x) that maps an input x to an output f(x), then the inverse function f^(-1)(x) maps the output f(x) back to the input x. In this article, we will explore how to find the inverse function of a linear function, specifically the function f(x) = 4(x+3).
What is a Linear Function?
A linear function is a function that can be written in the form f(x) = mx + b, where m is the slope of the function and b is the y-intercept. The graph of a linear function is a straight line. In the case of the function f(x) = 4(x+3), the slope is 4 and the y-intercept is 12.
Finding the Inverse Function
To find the inverse function of a linear function, we need to follow these steps:
- Swap the x and y variables: The first step in finding the inverse function is to swap the x and y variables. This means that we will replace x with y and y with x.
- Solve for y: Once we have swapped the x and y variables, we need to solve for y. This involves isolating y on one side of the equation.
- Replace y with x: Finally, we need to replace y with x to get the inverse function.
Finding the Inverse of f(x) = 4(x+3)
Let's apply the steps above to find the inverse function of f(x) = 4(x+3).
Step 1: Swap the x and y variables
We start by swapping the x and y variables. This gives us:
x = 4(y+3)
Step 2: Solve for y
Next, we need to solve for y. To do this, we will isolate y on one side of the equation. We can do this by subtracting 12 from both sides of the equation and then dividing both sides by 4.
x - 12 = 4y (x - 12) / 4 = y
Step 3: Replace y with x
Finally, we need to replace y with x to get the inverse function.
f^(-1)(x) = (x - 12) / 4
Simplifying the Inverse Function
We can simplify the inverse function by combining the constants in the numerator.
f^(-1)(x) = (x - 12) / 4 f^(-1)(x) = (x - 34) / 4 f^(-1)(x) = (x - 34) / 4 f^(-1)(x) = (x - 12) / 4
Conclusion
In this article, we have seen how to find the inverse function of a linear function, specifically the function f(x) = 4(x+3). We followed the steps of swapping the x and y variables, solving for y, and replacing y with x to get the inverse function. The inverse function of f(x) = 4(x+3) is f^(-1)(x) = (x - 12) / 4.
Answer
The correct answer is:
Introduction
In the previous article, we explored how to find the inverse function of a linear function, specifically the function f(x) = 4(x+3). In this article, we will answer some frequently asked questions about inverse functions.
Q: What is the purpose of finding the inverse function?
A: The purpose of finding the inverse function is to reverse the operation of another function. In other words, if we have a function f(x) that maps an input x to an output f(x), then the inverse function f^(-1)(x) maps the output f(x) back to the input x.
Q: How do I know if a function has an inverse?
A: A function has an inverse if it is one-to-one, meaning that each output value corresponds to exactly one input value. In other words, if a function passes the horizontal line test, then it has an inverse.
Q: What is the difference between a function and its inverse?
A: A function and its inverse are two different functions that "undo" each other. In other words, if we apply a function to an input, then apply its inverse to the output, we get back the original input.
Q: Can a function have more than one inverse?
A: No, a function cannot have more than one inverse. The inverse of a function is unique, meaning that there is only one function that reverses the operation of the original function.
Q: How do I find the inverse of a function that is not linear?
A: Finding the inverse of a function that is not linear can be more challenging than finding the inverse of a linear function. However, there are some general techniques that can be used, such as:
- Graphing: Graphing the function and its inverse can help to visualize the relationship between the two functions.
- Algebraic manipulation: Using algebraic manipulation, such as substitution and elimination, can help to find the inverse of a function.
- Using a calculator: Using a calculator, such as a graphing calculator, can help to find the inverse of a function.
Q: Can a function have an inverse if it is not one-to-one?
A: No, a function cannot have an inverse if it is not one-to-one. The inverse of a function is only defined for one-to-one functions.
Q: What is the relationship between a function and its inverse?
A: The relationship between a function and its inverse is that they "undo" each other. In other words, if we apply a function to an input, then apply its inverse to the output, we get back the original input.
Q: Can a function have an inverse if it is not continuous?
A: No, a function cannot have an inverse if it is not continuous. The inverse of a function is only defined for continuous functions.
Conclusion
In this article, we have answered some frequently asked questions about inverse functions. We have discussed the purpose of finding the inverse function, how to know if a function has an inverse, and the difference between a function and its inverse. We have also discussed how to find the inverse of a function that is not linear and the relationship between a function and its inverse.
Answer Key
- Q: What is the purpose of finding the inverse function? A: The purpose of finding the inverse function is to reverse the operation of another function.
- Q: How do I know if a function has an inverse? A: A function has an inverse if it is one-to-one, meaning that each output value corresponds to exactly one input value.
- Q: What is the difference between a function and its inverse? A: A function and its inverse are two different functions that "undo" each other.
- Q: Can a function have more than one inverse? A: No, a function cannot have more than one inverse.
- Q: How do I find the inverse of a function that is not linear? A: Finding the inverse of a function that is not linear can be more challenging than finding the inverse of a linear function. However, there are some general techniques that can be used, such as graphing, algebraic manipulation, and using a calculator.
- Q: Can a function have an inverse if it is not one-to-one? A: No, a function cannot have an inverse if it is not one-to-one.
- Q: What is the relationship between a function and its inverse? A: The relationship between a function and its inverse is that they "undo" each other.
- Q: Can a function have an inverse if it is not continuous? A: No, a function cannot have an inverse if it is not continuous.