Problem 48
Question
In Exercises 43-48, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. \(f(x) = \frac{1}{8} (x+2)^2 - 1\)
Step-by-Step Solution
Verified Answer
The results of the graphing and the Horizontal Line Test will determine whether or not the function \(f(x) = \frac{1}{8} (x+2)^2 - 1\) is one-to-one and has an inverse function. Remember, a function is one-to-one if and only if any horizontal line intersects the graph at most once.
1Step 1: Graph the Function
Use any graphing utility you prefer to input the equation \[f(x) = \frac{1}{8} (x+2)^2 - 1\] and generate its graph.
2Step 2: Perform the Horizontal Line Test
The Horizontal Line Test involves drawing horizontal lines across the graph of the function. If any horizontal line intersects the graph more than once, the function fails the test and is not one-to-one.
3Step 3: Interpret the Results
Examine the graph and the results of the Horizontal Line Test. If the function is one-to-one, it will pass the Horizontal Line Test and thus has an inverse. If it is not, then it doesn’t have an inverse in its current domain.
Key Concepts
Graphing UtilityOne-to-One FunctionsInverse FunctionsQuadratic Functions
Graphing Utility
A graphing utility is an invaluable tool for students and educators alike when it comes to visualizing mathematical functions. Whether it's an app on your smartphone, a program on your computer, or an advanced calculator, a graphing utility takes an equation and produces the corresponding graph. This visual representation allows for a deeper understanding of the function’s behavior.
In using a graphing utility to graph the function \(f(x) = \frac{1}{8} (x+2)^2 - 1\), you simplify the process of plotting countless points manually on a graph. After inputting the equation, the utility will swiftly display the curve, highlighting crucial features such as the vertex, the axis of symmetry, and the direction in which the parabola opens.
This instant feedback is essential for conducting tests like the Horizontal Line Test to determine if the function possesses certain characteristics, such as being one-to-one, which is a pre-condition for the existence of an inverse function.
In using a graphing utility to graph the function \(f(x) = \frac{1}{8} (x+2)^2 - 1\), you simplify the process of plotting countless points manually on a graph. After inputting the equation, the utility will swiftly display the curve, highlighting crucial features such as the vertex, the axis of symmetry, and the direction in which the parabola opens.
This instant feedback is essential for conducting tests like the Horizontal Line Test to determine if the function possesses certain characteristics, such as being one-to-one, which is a pre-condition for the existence of an inverse function.
One-to-One Functions
One-to-one functions, in the world of mathematics, have a special significance because each element in the domain maps to a unique element in the range. In other words, no two different inputs produce the same output. This concept is pivotal when determining if a function has an inverse.
The Horizontal Line Test is a simple graphical check to see if a function is one-to-one. If any horizontal line drawn through the graph of the function crosses the graph at more than one point, the function fails the test and is not one-to-one. For the given function \(f(x) = \frac{1}{8} (x+2)^2 - 1\), when we apply the Horizontal Line Test, it is evident that the test will fail since it is a quadratic function, and by definition, quadratic functions are not one-to-one.
Understanding this principle is crucial, as one-to-one functions maintain a one-to-one correspondence between the elements of their domains and ranges, which is a necessary condition for the existence of an inverse function.
The Horizontal Line Test is a simple graphical check to see if a function is one-to-one. If any horizontal line drawn through the graph of the function crosses the graph at more than one point, the function fails the test and is not one-to-one. For the given function \(f(x) = \frac{1}{8} (x+2)^2 - 1\), when we apply the Horizontal Line Test, it is evident that the test will fail since it is a quadratic function, and by definition, quadratic functions are not one-to-one.
Understanding this principle is crucial, as one-to-one functions maintain a one-to-one correspondence between the elements of their domains and ranges, which is a necessary condition for the existence of an inverse function.
Inverse Functions
Inverse functions are like mathematical mirrors. For a function \(f\), its inverse \(f^{-1}\) reverses the roles of inputs and outputs, meaning it takes the output of \(f\) and produces the original input. For \(f\) to have an inverse \(f^{-1}\), \(f\) must be one-to-one to ensure that each output comes from one unique input.
When applying the Horizontal Line Test to \(f(x) = \frac{1}{8} (x+2)^2 - 1\), we see that \(f\) does not satisfy the one-to-one condition and thus does not have an inverse function over its entire domain. However, by restricting the domain appropriately, we can make \(f\) one-to-one and then find an inverse. This is often done in practice to explore inverse relationships, even for functions that are not one-to-one across their entire domain.
When applying the Horizontal Line Test to \(f(x) = \frac{1}{8} (x+2)^2 - 1\), we see that \(f\) does not satisfy the one-to-one condition and thus does not have an inverse function over its entire domain. However, by restricting the domain appropriately, we can make \(f\) one-to-one and then find an inverse. This is often done in practice to explore inverse relationships, even for functions that are not one-to-one across their entire domain.
Quadratic Functions
Quadratic functions form parabolas when graphed and are represented by the general equation \(ax^2 + bx + c\). The quadratic function provided, \(f(x) = \frac{1}{8} (x+2)^2 - 1\), is a specific type of quadratic where the parabola opens upward (as the coefficient of \(x^2\) is positive) and is shifted horizontally and vertically from the origin.
Due to their symmetrical properties, quadratic functions do not pass the Horizontal Line Test except when the domain is restricted to making them one-to-one. In other words, over their full domain, they are not invertible. This property is seen in the function \(f(x)\) provided, where any horizontal line above the vertex will intersect the parabola at two points, indicating that the function is not one-to-one. Hence, in their natural form, quadratics do not possess inverse functions because they cannot meet the one-to-one criteria.
Due to their symmetrical properties, quadratic functions do not pass the Horizontal Line Test except when the domain is restricted to making them one-to-one. In other words, over their full domain, they are not invertible. This property is seen in the function \(f(x)\) provided, where any horizontal line above the vertex will intersect the parabola at two points, indicating that the function is not one-to-one. Hence, in their natural form, quadratics do not possess inverse functions because they cannot meet the one-to-one criteria.
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