Problem 43
Question
Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$g(x)=-x^{2}+8, x \geq 0$$
Step-by-Step Solution
Verified Answer
The inverse of the function \(g(x)=-x^{2}+8, x \geq 0\) is \(g^{-1}(x) = \sqrt{x-8}\), \(x \geq 8\). When graphed on the same set of axes, \(g(x)\) is a rightward facing half of a downward parabola, and \(g^{-1}(x)\) is the right half of an upward facing parabola.
1Step 1: Switch x and y
To find the inverse of the function \(g(x)=-x^{2}+8, x \geq 0\), the roles of x and y must be switched. This results in the equation: \(x=-y^{2}+8\).
2Step 2: Solve for y
The next step is to solve the equation for \(y\). First, subtract \(8-x\) from both sides to get \(-y^2 = 8-x\). Multiply by -1 to get \(y^2 = x-8\). Taking the square root of both sides results in \(y=\sqrt{x-8}\). Note that the domain is \(x\geq 8\), otherwise, would get complex results.
3Step 3: Graph the functions
The next step is graphing the original function (\(g(x)\)) and its inverse (\(g^{-1}(x)\)) on the same set of axes. \(g(x)=-x^{2}+8\) is a downward facing parabola with a vertex at (0,8), and (\(g^{-1}(x)\) = \(\sqrt{x-8}\)) is the right half of an upward facing parabola with the vertex at (8,0). Note that due to the restriction of the original function to \(x\geq 0\), only the right half of the original parabola should be graphed on the same axes.
Key Concepts
Graphing FunctionsQuadratic FunctionsDomain and Range
Graphing Functions
Graphing functions is a powerful way to visualize mathematical relationships.
By translating a function into a graph, we can understand how variables interact and see patterns more clearly. One important aspect of graphing is the identification of key features like the vertex, intercepts, and the overall shape of the graph.
For our function, the graph of the original function, \(g(x) = -x^2 + 8\), forms a downward-facing parabola. The vertex, which is the highest point of this graph due to the negative coefficient of \(x^2\), is located at (0, 8). Because the function is restricted to \(x \geq 0\), only the right half of this parabola is plotted.
The inverse function, \(y = \sqrt{x - 8}\), represents a transformation of this shape. Its graph is the right half of an upward-facing parabola, and its vertex appears at (8, 0). Both functions on the same graph provide a mirror-like interaction across the line \(y=x\).
By translating a function into a graph, we can understand how variables interact and see patterns more clearly. One important aspect of graphing is the identification of key features like the vertex, intercepts, and the overall shape of the graph.
For our function, the graph of the original function, \(g(x) = -x^2 + 8\), forms a downward-facing parabola. The vertex, which is the highest point of this graph due to the negative coefficient of \(x^2\), is located at (0, 8). Because the function is restricted to \(x \geq 0\), only the right half of this parabola is plotted.
The inverse function, \(y = \sqrt{x - 8}\), represents a transformation of this shape. Its graph is the right half of an upward-facing parabola, and its vertex appears at (8, 0). Both functions on the same graph provide a mirror-like interaction across the line \(y=x\).
- Downward vs. upward parabola shape.
- Vertex locations differ.
Quadratic Functions
Quadratic functions typically take the form \(ax^2 + bx + c\). In our exercise, \(g(x) = -x^2 + 8\), there is no \(bx\) term, simplifying our focus. Quadratic functions are known for their U-shaped graphs called parabolas.
There are a few key characteristics of quadratic functions:
Only the section of the graph that passes the horizontal line test can have an inverse that is a function.
This is why the original function \(g(x)\) has the restriction \(x \geq 0\). Without this restriction, \(g(x)\) would not be one-to-one and would not have an inverse that is a function.
There are a few key characteristics of quadratic functions:
- Direction of the parabola: determined by the leading coefficient (here it's -1, so the parabola opens downwards).
- The vertex: the turning point of the parabola (at (0, 8) in the given function).
- Symmetry: each parabola is symmetric about its vertical axis through the vertex.
Only the section of the graph that passes the horizontal line test can have an inverse that is a function.
This is why the original function \(g(x)\) has the restriction \(x \geq 0\). Without this restriction, \(g(x)\) would not be one-to-one and would not have an inverse that is a function.
Domain and Range
Understanding the domain and range of a function is crucial when grappling with inverse functions.
The domain consists of all possible \(x\) values a function can accept, while the range consists of all possible \(y\) values the function can produce.
For the given function, \(g(x) = -x^2 + 8\), the domain is restricted to \(x \geq 0\), ensuring the function remains one-to-one enough for its inverse to exist. The range, based on the vertex, is all real numbers less than or equal to 8.
In the inverse function, \(g^{-1}(x) = \sqrt{x-8}\), the new domain arises from the original range, becoming \(x \geq 8\). Consequently, the range now becomes \(y \geq 0\), reflecting the original domain.
The domain consists of all possible \(x\) values a function can accept, while the range consists of all possible \(y\) values the function can produce.
For the given function, \(g(x) = -x^2 + 8\), the domain is restricted to \(x \geq 0\), ensuring the function remains one-to-one enough for its inverse to exist. The range, based on the vertex, is all real numbers less than or equal to 8.
In the inverse function, \(g^{-1}(x) = \sqrt{x-8}\), the new domain arises from the original range, becoming \(x \geq 8\). Consequently, the range now becomes \(y \geq 0\), reflecting the original domain.
- Original domain: \(x \geq 0\)
- Original range: \(y \leq 8\)
- Inverse domain: \(x \geq 8\)
- Inverse range: \(y \geq 0\)
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