Determine Whether The Function Has An Inverse Function.[q(x) = (x-9)^2]A. Yes, Q(x) Does Have An Inverse.B. No, Q(x) Does Not Have An Inverse.If It Does, Find The Inverse Function. (If An Answer Does Not Exist, Enter DNE.)[q^{-1}(x) = \square]
Determining the Existence and Finding the Inverse of a Function
In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). The concept of an inverse function is crucial in understanding the properties of functions and their behavior. In this article, we will explore the existence and finding of the inverse of a given function, specifically the quadratic function q(x) = (x-9)^2.
An inverse function is a function that reverses the operation of the original function. In other words, if we have a function f(x) and its inverse f^{-1}(x), then f(f^{-1}(x)) = x and f^{-1}(f(x)) = x. This means that the inverse function undoes the operation of the original function, and vice versa.
The given function is q(x) = (x-9)^2. To determine whether this function has an inverse, we need to examine its properties. Specifically, we need to check if the function is one-to-one (injective), which is a necessary condition for a function to have an inverse.
A function is one-to-one if it assigns distinct outputs to distinct inputs. In other words, if f(x) = f(y), then x = y. This means that a one-to-one function has a unique output for each input.
To determine if the given function q(x) = (x-9)^2 is one-to-one, we need to examine its graph. The graph of a quadratic function is a parabola that opens upward or downward. In this case, the graph of q(x) = (x-9)^2 is a parabola that opens upward.
The graph of q(x) = (x-9)^2 has a minimum point at x = 9, which is the vertex of the parabola. The graph is symmetric about the vertical line x = 9. This means that for every point (x, y) on the graph, there is a corresponding point (9-x, y) on the graph.
Based on the analysis of the graph, we can conclude that the given function q(x) = (x-9)^2 is not one-to-one. This is because the graph is symmetric about the vertical line x = 9, which means that there are two distinct inputs (x and 9-x) that produce the same output y.
Since the given function q(x) = (x-9)^2 is not one-to-one, it does not have an inverse function. This is because a function must be one-to-one in order to have an inverse.
However, if we restrict the domain of the function to the interval x ≥ 9, then the function becomes one-to-one. In this case, we can find the inverse function by interchanging the x and y variables and solving for y.
To find the inverse function, we start by interchanging the x and y variables:
x = (y-9)^2
Next, we solve for y:
y = ±√(x+9)
Since the domain of the function is restricted to x ≥ 9, we take the positive square root:
y = √(x+9)
The inverse function of q(x) = (x-9)^2 is q^{-1}(x) = √(x+9).
In conclusion, the given function q(x) = (x-9)^2 does not have an inverse function. However, if we restrict the domain of the function to the interval x ≥ 9, then the function becomes one-to-one, and we can find the inverse function by interchanging the x and y variables and solving for y.
The final answer is:
A. No, q(x) does not have an inverse.
If it does, find the inverse function:
q^{-1}(x) = √(x+9)
Q&A: Determining the Existence and Finding the Inverse of a Function
In our previous article, we explored the concept of inverse functions and determined whether the given function q(x) = (x-9)^2 has an inverse. We also found the inverse function by restricting the domain of the function to the interval x ≥ 9. In this article, we will answer some frequently asked questions related to determining the existence and finding the inverse of a function.
A: A function must be one-to-one (injective) in order to have an inverse. This means that the function must assign distinct outputs to distinct inputs.
A: To determine if a function is one-to-one, you can examine its graph. If the graph is symmetric about a vertical line, then the function is not one-to-one. You can also use the horizontal line test: if a horizontal line intersects the graph at more than one point, then the function is not one-to-one.
A: The horizontal line test is a method used to determine if a function is one-to-one. If a horizontal line intersects the graph of a function at more than one point, then the function is not one-to-one.
A: To find the inverse of a function, you can interchange the x and y variables and solve for y. This will give you the inverse function.
A: If the function is not one-to-one, then it does not have an inverse. However, you can try restricting the domain of the function to a specific interval, and then find the inverse function.
A: No, a function can only have one inverse. However, if the function is not one-to-one, then you can try restricting the domain of the function to a specific interval, and then find the inverse function.
A: To check if the inverse function is correct, you can plug in the original function and the inverse function into the equation f(f^{-1}(x)) = x and f^{-1}(f(x)) = x. If both equations are true, then the inverse function is correct.
A: If the inverse function is not a function, then it is not a valid inverse. In this case, you can try restricting the domain of the function to a specific interval, and then find the inverse function.
A: No, a function must be continuous in order to have an inverse. If the function is not continuous, then it does not have an inverse.
In conclusion, determining the existence and finding the inverse of a function requires careful analysis of the function's properties. By using the horizontal line test and interchanging the x and y variables, you can determine if a function has an inverse and find the inverse function. Remember to check if the inverse function is correct by plugging it into the equation f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.
The final answer is:
A. No, q(x) does not have an inverse.
If it does, find the inverse function:
q^{-1}(x) = √(x+9)