Invertible Functions Which Function Has An Inverse That Is Also A Function

Which of the following functions has an inverse that is also a function? g(x)=2x-3, k(x)=-9x^2, f(x)=|x+2|, w(x)=-20

Determining which function has an inverse that is also a function involves understanding the concept of invertible functions and the horizontal line test. A function has an inverse that is also a function if and only if it is a one-to-one function. A one-to-one function, also known as an injective function, is a function where each element of the range is associated with exactly one element in the domain. Graphically, this means that no horizontal line intersects the graph of the function more than once. This is known as the horizontal line test. Let's examine the given functions:

Understanding Invertible Functions and the Horizontal Line Test

To effectively identify which function possesses an inverse that is also a function, a solid grasp of invertible functions and the horizontal line test is crucial. An invertible function is essentially a function that has an inverse which is also a function. This property hinges on the function being one-to-one, meaning each element in the function's range corresponds to only one element in its domain. This one-to-one correspondence ensures that when we reverse the roles of the input and output to find the inverse, we still maintain a valid function.

The horizontal line test provides a visual method to determine if a function is one-to-one. If any horizontal line intersects the graph of the function at more than one point, the function fails the test and is not one-to-one. This is because the points of intersection represent different input values (x-values) that produce the same output value (y-value), violating the one-to-one principle. Conversely, if no horizontal line intersects the graph more than once, the function passes the horizontal line test and is considered one-to-one.

Understanding the horizontal line test is crucial to determine if a function has an inverse. A function has an inverse function if it is a one-to-one or injective function. Graphically, a function is one-to-one if no horizontal line intersects the graph more than once. This is because each $y$-value must correspond to a unique $x$-value. In simpler terms, if we draw a horizontal line across the graph, it should only intersect the graph at most once for the inverse to be a function. Failing this test indicates that the original function has multiple $x$-values mapping to the same $y$-value, which means its inverse would not be a function. Therefore, functions that pass the horizontal line test are guaranteed to have inverses that are also functions, making this test a pivotal tool in identifying invertible functions.

Let's consider a few examples to illustrate the application of the horizontal line test. Take, for instance, a linear function like $f(x) = 2x + 1$. Its graph is a straight line, and any horizontal line drawn will intersect it only once. Therefore, this function is one-to-one and has an inverse function. On the other hand, consider a quadratic function like $g(x) = x^2$. Its graph is a parabola, and any horizontal line above the vertex will intersect it twice. This means the function is not one-to-one, and its inverse is not a function unless we restrict the domain. The horizontal line test is an indispensable tool for quickly assessing whether a function has an inverse that is also a function, saving time and ensuring accuracy in mathematical analysis.

Analyzing the Given Functions

1. $g(x) = 2x - 3$

This is a linear function. Linear functions, except for horizontal lines, are always one-to-one. To confirm, we can apply the horizontal line test. Imagine drawing a horizontal line across the graph of $g(x)$. It will only intersect the line at one point. Therefore, $g(x)$ is a one-to-one function and has an inverse that is also a function.

Consider the function $g(x) = 2x - 3$. This is a linear function with a slope of 2 and a y-intercept of -3. The graph of this function is a straight line that slopes upwards from left to right. To determine if this function has an inverse that is also a function, we can apply the horizontal line test. If we draw any horizontal line across the graph of $g(x)$, it will intersect the line at exactly one point. This confirms that $g(x)$ is a one-to-one function, which means that each y-value corresponds to a unique x-value. Since $g(x)$ passes the horizontal line test, it has an inverse function. To find the inverse, we can switch $x$ and $y$ and solve for $y$: $x = 2y - 3$, which gives us $y = (x + 3) / 2$. The inverse function, $g^{-1}(x) = (x + 3) / 2$, is also a linear function, thus a function. This detailed analysis underscores that $g(x) = 2x - 3$ not only has an inverse but also confirms that the inverse is, in fact, a function.

To further illustrate the concept, let's consider two different $x$-values, say $x_1$ and $x_2$, where $x_1 e x_2$. For $g(x)$ to be one-to-one, it must be the case that $g(x_1) e g(x_2)$. Let's assume $g(x_1) = g(x_2)$. Then, $2x_1 - 3 = 2x_2 - 3$. Adding 3 to both sides gives $2x_1 = 2x_2$, and dividing by 2 yields $x_1 = x_2$. This contradiction proves that if $g(x_1) = g(x_2)$, then $x_1$ must equal $x_2$, demonstrating that $g(x)$ is indeed a one-to-one function. This algebraic proof complements the graphical horizontal line test, providing a comprehensive understanding of why $g(x) = 2x - 3$ possesses an inverse that is also a function.

2. $k(x) = -9x^2$

This is a quadratic function. Quadratic functions form parabolas, which fail the horizontal line test. A horizontal line can intersect the parabola at two points (except at the vertex). For example, both $x = 1$ and $x = -1$ give $k(x) = -9$. Therefore, $k(x)$ does not have an inverse that is a function.

Delving deeper into the analysis of $k(x) = -9x^2$, it is essential to recognize that this function represents a parabola that opens downwards due to the negative coefficient in front of the $x^2$ term. The vertex of this parabola is at the origin (0,0), and the axis of symmetry is the y-axis. To determine whether $k(x)$ has an inverse that is also a function, we apply the horizontal line test. If we draw a horizontal line, say $y = -9$, it intersects the parabola at two points, $x = 1$ and $x = -1$. This means that two different $x$-values produce the same $y$-value, specifically $k(1) = -9(1)^2 = -9$ and $k(-1) = -9(-1)^2 = -9$. This observation confirms that $k(x)$ is not a one-to-one function, thereby failing the condition necessary for its inverse to be a function.

To rigorously demonstrate that $k(x)$ does not have an inverse function, consider the definition of a one-to-one function. A function is one-to-one if for every $y$-value in the range, there is at most one $x$-value in the domain that maps to it. In this case, we have shown that for $y = -9$, there are two $x$-values, $x = 1$ and $x = -1$, that satisfy $k(x) = y$. This directly contradicts the one-to-one requirement. Another way to look at this is to attempt to find the inverse algebraically. If we set $y = -9x^2$ and try to solve for $x$, we get $x^2 = -y/9$, so $x = \\\pm\\\sqrt{-y/9}$. The presence of the $\\\\pm$ sign indicates that there are two possible $x$-values for each $y$-value (except for $y = 0$), further confirming that the inverse is not a function without restricting the domain. Therefore, $k(x) = -9x^2$ does not have an inverse that is also a function over its entire domain.

3. $f(x) = |x + 2|$

This is an absolute value function. Absolute value functions form a V-shape, which also fails the horizontal line test. For example, both $x = 0$ and $x = -4$ give $f(x) = 2$. Therefore, $f(x)$ does not have an inverse that is a function.

The function $f(x) = |x + 2|$ is an absolute value function, which graphically presents as a V-shaped figure. The vertex of this V-shape is located at the point $(-2, 0)$, where the function changes direction. To ascertain if $f(x)$ has an inverse that is also a function, we once again employ the horizontal line test. Imagine drawing a horizontal line above the vertex, say at $y = 2$. This line will intersect the graph of $f(x)$ at two points. Specifically, $f(0) = |0 + 2| = 2$ and $f(-4) = |-4 + 2| = |-2| = 2$. The existence of two distinct $x$-values producing the same $y$-value immediately indicates that $f(x)$ fails the horizontal line test and is not a one-to-one function. Consequently, the inverse of $f(x)$ is not a function.

To provide a more rigorous explanation, we can consider the definition of the absolute value function itself. The absolute value function, $|x|$, returns the non-negative magnitude of $x$. Thus, $|x|$ is $x$ if $x ext{ is not less than } 0$, and $-x$ if $x < 0$. Similarly, $|x + 2|$ is $x + 2$ if $x + 2 ext{ is not less than } 0$, and $-(x + 2)$ if $x + 2 < 0$. This behavior creates the symmetry characteristic of the V-shape, which leads to the failure of the horizontal line test. If we attempt to find an inverse algebraically, we would encounter the same issue. Setting $y = |x + 2|$ and solving for $x$ requires considering two cases: $x + 2 = y$ and $-(x + 2) = y$. These cases lead to two potential solutions for $x$, further demonstrating that the inverse is not a function without restricting the domain. Therefore, $f(x) = |x + 2|$ does not have an inverse that is a function over its entire domain due to its symmetrical nature around the vertex.

4. $w(x) = -20$

This is a constant function. Constant functions are horizontal lines. A horizontal line fails the horizontal line test because any horizontal line (including itself) will intersect it infinitely many times. Therefore, $w(x)$ does not have an inverse that is a function.

Consider the function $w(x) = -20$. This function is a constant function, meaning that for every input $x$, the output is always -20. Graphically, this is represented by a horizontal line at $y = -20$. To determine if $w(x)$ has an inverse that is also a function, we apply the horizontal line test. In this scenario, the graph of the function is a horizontal line. Therefore, any horizontal line at $y = -20$ will intersect the graph infinitely many times. This clearly demonstrates that $w(x)$ is not a one-to-one function. In a constant function, every input maps to the same output, making it impossible to uniquely reverse the mapping, which is necessary for the inverse to be a function.

To elaborate on why constant functions do not have inverses that are functions, consider the definition of an inverse function. The inverse function should reverse the mapping of the original function. In the case of $w(x) = -20$, every real number maps to -20. If we try to create an inverse, we would need to map -20 back to a single, unique input. However, since infinitely many inputs map to -20, there is no single, unique way to reverse this mapping. Consequently, the inverse cannot be a function because a function must have a unique output for each input. Algebraically, if we set $y = -20$ and try to solve for $x$, we find that $x$ can be any real number. This algebraic perspective reinforces the graphical understanding that $w(x) = -20$ does not have an inverse function. Therefore, the constant nature of $w(x) = -20$ prevents it from having an inverse that is also a function.

Conclusion

Only $g(x) = 2x - 3$ has an inverse that is also a function because it is a linear function and passes the horizontal line test. The other functions, $k(x)$, $f(x)$, and $w(x)$, do not have inverses that are functions because they fail the horizontal line test.

In summary, determining whether a function has an inverse that is also a function involves understanding the concept of one-to-one functions and applying the horizontal line test. Linear functions, excluding horizontal lines, generally have inverses that are functions, while quadratic, absolute value, and constant functions do not, unless their domains are restricted to make them one-to-one. Understanding these principles allows for quick and accurate assessment of function invertibility.