Question

Find the inverse of the given function fand verify that \(f\left(f^{-1}(x)\right)=x\) for all \(x\) in the domain of \(f^{-1}\), and \(50^{-1}(f(x))=x\) for all \(x\) in the domain off. $$ f(x)=3+10^{x} $$

Short answer

The inverse of \( f(x) = 3 + 10^x \) is \( f^{-1}(x) = \log_{10}(x - 3) \). The verification shows the inverse is correct.

Step-by-step solution

Write the equation for the inverse function

The given function is \( f(x) = 3 + 10^x \). To find the inverse, we start by setting \( y = f(x) \), which gives us \( y = 3 + 10^x \).

Solve for x in terms of y

Rearrange the equation \( y = 3 + 10^x \) to solve for \( x \). Subtract 3 from both sides to get \( y - 3 = 10^x \).

Apply logarithm to isolate x

To isolate \( x \), take the logarithm base 10 of both sides: \( \log_{10}(y - 3) = x \). Thus, the inverse function \( f^{-1}(x) \) is \( \log_{10}(x - 3) \).

Verify the inverse using composition one way

Confirm \( f(f^{-1}(x)) = x \): Substitute \( f^{-1}(x) = \log_{10}(x - 3) \) into \( f \). This gives \( f(\log_{10}(x - 3)) = 3 + 10^{\log_{10}(x - 3)} = 3 + (x - 3) = x \). Since both sides equal \( x \), the first verification is complete.

Verify the inverse using composition the other way

Confirm \( f^{-1}(f(x)) = x \): Substitute \( f(x) = 3 + 10^x \) into \( f^{-1} \). This gives \( f^{-1}(3 + 10^x) = \log_{10}((3 + 10^x) - 3) = \log_{10}(10^x) = x \). Since both sides equal \( x \), the second verification is complete.

Key concepts

Logarithmic Function

Logarithmic functions are the inverse of exponential functions. In the context of our problem, we deal with the function \( f(x) = 3 + 10^x \), where the base of the exponential part is 10, indicating we will use the base-10 logarithm (also known as the common logarithm) to find its inverse. The general principle is to "undo" the effect of the exponential function, which in this case is achieved by applying the logarithm.

Here’s the process:

• Start by expressing the exponential function in terms of its inverse: if \( y = 10^x \), then \( x = \log_{10}(y) \).
• Apply this to isolate \( x \) in our function, \( f(x) = 3 + 10^x \). Begin with \( y = 3 + 10^x \).
• Rearrange it to \( y - 3 = 10^x \), and then take the logarithm of both sides to get \( x = \log_{10}(y - 3) \).

This yields the inverse function \( f^{-1}(x) = \log_{10}(x - 3) \). Understanding how logarithms "reverse" exponentiation is central to correctly determining the inverse of exponential functions.

Function Composition

Function composition involves combining two functions to form a new function, where the output of one function becomes the input of the other. In the context of finding and verifying inverses, composition helps in confirming that two functions are indeed inverses.

For our specific example, when we composed \( f \) and \( f^{-1} \), we performed:

• \( f(f^{-1}(x)) = f(\log_{10}(x - 3)) = 3 + 10^{\log_{10}(x - 3)} \)
• Given that \( 10^{\log_{10}(x-3)} = x-3 \), this simplifies to \( x \), proving the composition is correct.

Similarly, composing in the reverse order:

• \( f^{-1}(f(x)) = \log_{10}((3 + 10^x) - 3) = \log_{10}(10^x) \)
• Since \( \log_{10}(10^x) = x \), this simplifies to \( x \) as well.

The essence of this step is to show that each function undoes the effect of the other, reinforcing that they are true inverses.

Domain and Range

Understanding the domain and range is crucial when working with inverse functions. The domain of the original function becomes the range of the inverse and vice versa.

For our function \( f(x) = 3 + 10^x \), the domain is all real numbers because an exponential function with any real exponent is defined. The range of \( f(x) \) is \((3, \infty)\) as \( 10^x \) approaches \( \infty \) and the smallest value it can approach (excluding) is 3, thus adding 3 shifts the entire curve upwards.

For the inverse function \( f^{-1}(x) = \log_{10}(x - 3) \), the domain must be \((3, \infty)\) since \( x - 3 \) must be positive for the logarithm to be defined, and the range will be all real numbers, thus reflecting the domain of the original function.

Mastering these concepts helps predict the behavior of inverses accurately and ensure we apply the functions correctly within their valid intervals.

Verification of Functions

Verification ensures the accuracy of our inverse function calculations. We achieve this through function composition where each direction of composition must return the original input.

For our functions \( f(x) = 3 + 10^x \) and its inverse \( f^{-1}(x) = \log_{10}(x - 3) \), we verify by:

• Confirming that \( f(f^{-1}(x)) = x \). When applying \( f \) to \( f^{-1} \), it returns \( x \), indicating that \( f^{-1} \) correctly reverses \( f \).
• Checking \( f^{-1}(f(x)) = x \). When applying \( f^{-1} \) to \( f \), it also returns \( x \), indicating that \( f \) and \( f^{-1} \) are true inverses.

This double verification allows us to confidently assert the accuracy of the inverse function derived, ensuring it operates as intended over the specified domain and range.