Problem 45
Question
\(y=\frac{-1}{x^{2}}\)
Step-by-Step Solution
Verified Answer
The domain is all real numbers except zero, and the range is all negative real numbers.
1Step 1: Understand the Function
The given function is \(y = \frac{-1}{x^2}\). This is a reciprocal function where the numerator is -1 and the denominator is \(x^2\). This type of function is known for having vertical asymptotes and a hyperbola-shaped graph.
2Step 2: Find the Domain
The domain of the function is the set of all possible values of \(x\) for which the function is defined. Since division by zero is undefined, \(x^2\) cannot be zero. Thus, the domain is all real numbers except \(x = 0\), expressed as \(x \in \mathbb{R} \setminus \{0\}\).
3Step 3: Determine the Range
Since \(y = \frac{-1}{x^2}\), the function will only produce negative values because \(x^2 \) is always positive, and the numerator is -1. Therefore, the range is all negative real numbers \(y < 0\).
4Step 4: Analyze Asymptotes
There is a vertical asymptote at \(x = 0\) because the function is undefined there, and a horizontal asymptote at \(y = 0\) because as \(x\) approaches infinity or negative infinity, \(y\) approaches 0.
5Step 5: Identify Symmetry
The function is symmetric with respect to the y-axis. This is evident because if you replace \(x\) with \(-x\), the function remains unchanged: \(y = \frac{-1}{(-x)^2} = \frac{-1}{x^2}\).
Key Concepts
Domain and RangeVertical and Horizontal AsymptotesFunction Symmetry
Domain and Range
The domain of a reciprocal function like \( y = \frac{-1}{x^2} \)is the set of all permissible values of \( x \) for which the function is defined. In this case, we must remember that division by zero is not allowed, so \( x^2 \)must never be zero. This means the domain of our function is all real numbers except \( x = 0 \), written as \( x \in \mathbb{R} \setminus \{0\} \).
Moving on to the range, it refers to the set of all possible output values \( y \)can take. With \( y = \frac{-1}{x^2} \), the numerator, -1, ensures that y is always negative, while the denominator, \( x^2 \), is always positive except at zero, which is excluded. Therefore, the range is all negative real numbers, specifically \( y < 0 \).Remember, range and domain help us understand the behavior and limitations of the function.
Moving on to the range, it refers to the set of all possible output values \( y \)can take. With \( y = \frac{-1}{x^2} \), the numerator, -1, ensures that y is always negative, while the denominator, \( x^2 \), is always positive except at zero, which is excluded. Therefore, the range is all negative real numbers, specifically \( y < 0 \).Remember, range and domain help us understand the behavior and limitations of the function.
Vertical and Horizontal Asymptotes
Asymptotes are lines that a graph approaches but never actually meets. For the function \( y = \frac{-1}{x^2} \), we identify both vertical and horizontal asymptotes.
First, a vertical asymptote occurs at \( x = 0 \). This happens because the function becomes undefined when \( x^2 = 0 \), i.e., it cannot divide by zero. Thus, \( x = 0 \)creates a boundary line that the graph never crosses.
For the horizontal asymptote, as \( x \)moves towards infinity in either direction, the value of \( y \) approaches 0 but remains negative. Therefore, a horizontal asymptote is located at \( y = 0 \). This concept helps us visualize how the function behaves at extreme values of \( x \). The graph gets closer to both asymptotes but never actually touches them.
First, a vertical asymptote occurs at \( x = 0 \). This happens because the function becomes undefined when \( x^2 = 0 \), i.e., it cannot divide by zero. Thus, \( x = 0 \)creates a boundary line that the graph never crosses.
For the horizontal asymptote, as \( x \)moves towards infinity in either direction, the value of \( y \) approaches 0 but remains negative. Therefore, a horizontal asymptote is located at \( y = 0 \). This concept helps us visualize how the function behaves at extreme values of \( x \). The graph gets closer to both asymptotes but never actually touches them.
Function Symmetry
A function is considered symmetric if its graph looks the same on either side of a certain line, like the y-axis. \( y = \frac{-1}{x^2} \) is symmetric with respect to the y-axis. This symmetry means the function reflects across this axis.
To verify symmetry, you can substitute \( x \) with \( -x \)and check if the function remains unchanged. Here, replacing \( x \) with \( -x \) results in \( y = \frac{-1}{(-x)^2} = \frac{-1}{x^2} \),confirming y-axis symmetry.
This property indicates that the graph is mirrored on either side of the y-axis, providing a visual clue to the function's behavior. Recognizing symmetry helps simplify the analysis of function graphs.
To verify symmetry, you can substitute \( x \) with \( -x \)and check if the function remains unchanged. Here, replacing \( x \) with \( -x \) results in \( y = \frac{-1}{(-x)^2} = \frac{-1}{x^2} \),confirming y-axis symmetry.
This property indicates that the graph is mirrored on either side of the y-axis, providing a visual clue to the function's behavior. Recognizing symmetry helps simplify the analysis of function graphs.