Problem 64
Question
Determine algebraically whether the function is one-to-one. Verify your answer graphically. If the function is one-to-one, find its inverse. $$h(x)=\frac{4}{x^{2}}$$
Step-by-Step Solution
Verified Answer
The function \(h(x) = \frac{4}{{x^{2}}}\) is not one-to-one, so it does not have an inverse function.
1Step 1: Test for One-to-One
To check if function is one-to-one, apply the horizontal line test. If any horizontal line intersects the graph of the function in more than one place, the function is not one-to-one. In case of \(h(x) = \frac{4}{{x^{2}}}\), any horizontal line above the x-axis will intersect the graph in two places (one on each side of the y-axis), so this function is not one-to-one.
2Step 2: Finding the Inverse of Function
Because the function is not one-to-one, there's no way to find an inverse that would be a function.
3Step 3: Graphical Verification
A graphical representation of the function \(h(x) = \frac{4}{{x^{2}}}\) shows a curve that hallows towards the x-axis from both sides. Any horizontal line above the x-axis will intersect the graph in two places, which visually confirms this function is not one-to-one.
Key Concepts
Horizontal Line TestFunction GraphInverse FunctionAlgebraic Verification
Horizontal Line Test
The horizontal line test is a simple yet powerful tool to determine if a function is one-to-one. A function is considered one-to-one when each output value corresponds to exactly one input value. To perform this test, take any horizontal line and see how it intersects with the graph of the function. If the line crosses the graph at more than one point, the function is not one-to-one.
For the function in question, \( h(x)=\frac{4}{x^2} \), imagine drawing horizontal lines above the x-axis. Since any of these lines will intersect the graph at two different points on the opposite sides of the y-axis, this function fails the horizontal line test. Therefore, \( h(x) \) is not a one-to-one function.
For the function in question, \( h(x)=\frac{4}{x^2} \), imagine drawing horizontal lines above the x-axis. Since any of these lines will intersect the graph at two different points on the opposite sides of the y-axis, this function fails the horizontal line test. Therefore, \( h(x) \) is not a one-to-one function.
Function Graph
The graph of a function provides a visual method to analyze its characteristics. For \( h(x) = \frac{4}{x^2} \), the graph creates a curve with two branches, each moving away symmetrically from the y-axis. This behavior is typical for functions involving square inverses, where you can observe symmetry with respect to both the x- and y-axes.
When graphed, the function forms a shape akin to a mirrored parabola opening downward, but without actually crossing the x-axis. This visualization helps reinforce why the function is not one-to-one, as previously determined through the horizontal line test, further confirming the inability to find a proper inverse function.
When graphed, the function forms a shape akin to a mirrored parabola opening downward, but without actually crossing the x-axis. This visualization helps reinforce why the function is not one-to-one, as previously determined through the horizontal line test, further confirming the inability to find a proper inverse function.
Inverse Function
An inverse function is one that undoes the work of the original function. If you have a function \( f(x) \) and its inverse \( f^{-1}(x) \), applying the original followed by the inverse should lead you back to your starting number: \( f(f^{-1}(x))=x\).
However, an inverse exists only for one-to-one functions. Given that \( h(x)=\frac{4}{x^2} \) is not one-to-one, it lacks an inverse that qualifies as a function. While we could technically rearrange \( y=\frac{4}{x^2} \) to solve for \( x \), yield would not satisfy conditions for a functional inverse due to multiple values of \( x \) for a single \( y \).
This is a crucial point since any attempt to plot an inverse for such non-one-to-one functions would result in a many-to-one mapping, violating the inversibility criterion.
However, an inverse exists only for one-to-one functions. Given that \( h(x)=\frac{4}{x^2} \) is not one-to-one, it lacks an inverse that qualifies as a function. While we could technically rearrange \( y=\frac{4}{x^2} \) to solve for \( x \), yield would not satisfy conditions for a functional inverse due to multiple values of \( x \) for a single \( y \).
This is a crucial point since any attempt to plot an inverse for such non-one-to-one functions would result in a many-to-one mapping, violating the inversibility criterion.
Algebraic Verification
Algebraic verification involves using equations and algebraic manipulation to determine properties of a function, such as its invertibility or one-to-oneness. Begin by assessing if different input values yield unique output values. If not, then the function fails this test.
For \( h(x)=\frac{4}{x^2} \), inputting both \( x=2 \) and \( x=-2 \) results in the same output of \( 1 \). This duplicity in output for distinct inputs affirms, algebraically, that \( h(x) \) is not one-to-one. Consequently, it reflects the result of the horizontal line test.
Such algebraic verification supplements graphical analysis and serves as an additional method to confirm the non-one-to-one nature of the function.
For \( h(x)=\frac{4}{x^2} \), inputting both \( x=2 \) and \( x=-2 \) results in the same output of \( 1 \). This duplicity in output for distinct inputs affirms, algebraically, that \( h(x) \) is not one-to-one. Consequently, it reflects the result of the horizontal line test.
Such algebraic verification supplements graphical analysis and serves as an additional method to confirm the non-one-to-one nature of the function.
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