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Magazine Chapter 3: Problem 15
Determine whether the function is one-to-one. \(f(x)=\frac{1}{x^{2}}\)
Short Answer
Expert verified
The function \( f(x) = \frac{1}{x^2} \) is not one-to-one.
Step by step solution
01
Understanding the Function
The function given is \( f(x) = \frac{1}{x^2} \). It's important to identify the domain of this function. The function is undefined at \( x = 0 \), so the domain is all real numbers except zero, \( x eq 0 \). This function is continuous on its domain.
02
Criteria for One-to-One Function
A function is one-to-one if and only if every element in the range corresponds to exactly one element in the domain. Alternatively, a function is one-to-one if, for every \( a \) and \( b \) in the domain, \( f(a) = f(b) \) implies \( a = b \). This characteristics can also be checked using the Horizontal Line Test.
03
Apply Horizontal Line Test
The Horizontal Line Test states that a function is one-to-one if no horizontal line intersects the graph of the function more than once. For \( f(x) = \frac{1}{x^2} \), graph it briefly or visualize the graph. It appears like a downward opening parabola centered at the origin and it shows horizontal lines would intersect the curve more than once.
04
Analyze Algebraically
To check algebraically, assume \( f(a) = f(b) \) for some \( a \) and \( b \) in the domain:\[ \frac{1}{a^2} = \frac{1}{b^2} \] Rearrange to find \( a^2 = b^2 \), which implies \( a = b \) or \( a = -b \). Hence the function is not one-to-one since two different numbers, \( a \) and \( -a \), can have the same function value.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Horizontal Line Test
To determine whether a function is one-to-one, we can use the horizontal line test. This test involves drawing horizontal lines across the graph of a function. If any horizontal line intersects the graph at more than one point, the function is not one-to-one.
For the function \( f(x) = \frac{1}{x^2} \), when graphed, it appears like a downward opening parabola. This means that if you were to draw horizontal lines across this graph, they would cross the graph at two points for most values of \( y \).
For the function \( f(x) = \frac{1}{x^2} \), when graphed, it appears like a downward opening parabola. This means that if you were to draw horizontal lines across this graph, they would cross the graph at two points for most values of \( y \).
- If any horizontal line crosses a graph more than once, that function is not one-to-one.
- For every horizontal line test, examine intersections鈥攎ultiple intersections indicate the function is not unique per output.
Defining the Function Domain
The domain of a function consists of all the possible input values (\( x \)) for which the function is defined. It's crucial to identify which values \( x \) can take to avoid undefined expressions, such as division by zero.
In the case of \( f(x) = \frac{1}{x^2} \), we identify that this function is undefined at \( x = 0 \) because division by zero is not possible. Therefore, the domain of this function is all real numbers except zero, \( x eq 0 \).
In the case of \( f(x) = \frac{1}{x^2} \), we identify that this function is undefined at \( x = 0 \) because division by zero is not possible. Therefore, the domain of this function is all real numbers except zero, \( x eq 0 \).
- Always check for values of \( x \) that cause division by zero.
- The domain specifies where the function exists and provides context for analyzing its behavior.
Exploring the Function Range
The range of a function is the set of all possible output values (\( y \)). It's the set of values that \( f(x) \) can take when \( x \) varies over its domain.
For \( f(x) = \frac{1}{x^2} \), the function outputs only positive values because squares result in non-negative numbers and the reciprocal of non-negative numbers remains positive. Therefore, the range is all positive real numbers, \( y > 0 \).
For \( f(x) = \frac{1}{x^2} \), the function outputs only positive values because squares result in non-negative numbers and the reciprocal of non-negative numbers remains positive. Therefore, the range is all positive real numbers, \( y > 0 \).
- Observing the range helps determine if a function can produce the desired outcomes for certain inputs.
- For functions involving squared terms, the range often excludes negative values.
Using Algebraic Analysis for One-to-One Determination
Algebraic analysis allows us to use equations to determine if a function is one-to-one. For any two inputs \( a \) and \( b \) in the domain of a function, if \( f(a) = f(b) \), and it follows that \( a = b \), the function is one-to-one.
For \( f(x) = \frac{1}{x^2} \), setting \( f(a) = f(b) \):
\[ \frac{1}{a^2} = \frac{1}{b^2} \]
Upon cross multiplication, we get \( a^2 = b^2 \), which simplifies to \( a = b \) or \( a = -b \). This implies that two different values \( a \) and \( -a \) can produce the same output, showing that the function is not one-to-one.
For \( f(x) = \frac{1}{x^2} \), setting \( f(a) = f(b) \):
\[ \frac{1}{a^2} = \frac{1}{b^2} \]
Upon cross multiplication, we get \( a^2 = b^2 \), which simplifies to \( a = b \) or \( a = -b \). This implies that two different values \( a \) and \( -a \) can produce the same output, showing that the function is not one-to-one.
- Using algebra provides a concrete way to test for one-to-oneness.
- Recognize that identical outputs arising from distinct inputs disqualify a function as one-to-one.
