Question

In Exercises \(25-32,\) use parametric graphing to graph \(f, f^{-1},\) and \(y=x\) $$f(x)=2^{-x}$$

Short answer

The graph of the function \(f(x)=2^{-x}\) starts from the y-axis at 1 and continues to decrease as x increases. The inverse function \(f^{-1}(x)=-\log_2(x)\) starts from x=1 and increases as x increases. The identity graph \(y=x\) is a straight line passing through the origin with a slope of 1.

Step-by-step solution

Graphing the Function

To graph \(f(x)=2^{-x}\), plot a suitable range of x-values against their corresponding y-values computed using the function. For instance, when \(x=1, f(x)=2^{-1}=0.5\), when \(x=2, f(x)=2^{-2}=0.25\) and so on.

Finding the Inverse Function

The inverse of a function reflects the original function over the line \(y=x\). Thus to find \(f^{-1}(x)\), we replace \(f(x)\) with \(y\), swap \(x\) and \(y\) and solve for \(y\). So, \(f(x)=2^{-x}\) becomes \(y = 2^{-x}\), then \(2^x = y^{-1}\), making \(x = y^{-1}\) or \(f^{-1}(x) = -\log_2(x)\)

Graphing the Inverse Function

Now, graph the function \(f^{-1}(x) = -\log_2(x)\), not forgetting the range of x-values should match with those used in graphing \(f(x)\)

Graphing the Identity Function

Finally, graph the identity function \(y=x\), which is a straight line passing through the origin with a slope of 1.

Key concepts

Inverse Functions

Understanding inverse functions is essential for grasping how two functions can be related in a special and symmetric way. In essence, an inverse function essentially undoes the action of the original function. To find the inverse function, denoted as f^-1(x), we take the output of the original function and figure out what input would lead to that output.

For the function in the exercise, f(x) = 2^-x, we follow a specific set of steps to find its inverse. We start by writing y = 2^-x, then we interchange the roles of x and y, resulting in x = 2^-y. After that, we solve this equation for y to express the inverse. In this case, the inverse function is found to be f^-1(x) = -log₂(x), which represents the necessary exponent to raise 2 to in order to get x as the result.

It's crucial to note that not all functions have inverses, and for a function to have an inverse, it must be bijective (both injective and surjective). Additionally, graphical representation helps illustrate this relationship: any function and its inverse function are reflections of each other along the identity function y = x. This mirrors the concept that they 'undo' each other's operations.

Logarithmic Functions

The logarithmic functions are the inverses of exponential functions and play a pivotal role in many areas of mathematics, including calculus. When dealing with the inverse of an exponential function, such as f(x) = 2^-x from our exercise, we enter the realm of logarithms.

A logarithm answers the question: 'To what exponent do we need to raise a certain base to obtain a given number?'. In the context of the provided exercise, the inverse function f^-1(x) = -log₂(x) asks us to find the exponent that 2 must be raised to in order to yield x. This is why the base of the logarithm matches the base of the exponential function it's derived from.

The properties of logarithmic functions reflect their relationship with exponentials. They allow us to solve exponential equations, model real-life phenomena such as sound intensity and earthquake magnitudes, and even simplify complex multiplication and division calculations into addition and subtraction problems when the numbers involved are powers of the same base.

Identity Function

The concept of the identity function is elegantly simple yet profoundly important in mathematics. An identity function is a function that always returns the same value that was used as its input; that is, for every x, f(x) = x.

In the context of graphing, the identity function y = x is represented by a straight line that cuts through the origin (0,0) at a 45-degree angle to both the x-axis and the y-axis. This line is the graphical equivalent of a mathematical mirror because it reflects the function across itself to produce the inverse function. As illustrated in Step 4 of the exercise solution, graphing the identity function alongside the function and its inverse function helps validate the accuracy of the inverse by checking if the function and its inverse are mirror images along y = x.

The identity function is also a fundamental concept in algebra because it's used to describe the behavior of elements that remain unchanged under a given operation. It serves as a benchmark for considering transformations, and in the relationship between a function and its inverse, where the composition of the two should yield the identity function for all values in their domain.