Problem 89 - Algebra A Combined Function

Question

Let \(f(x)=\log _{5} x\). Then \(g(x)=5^{x}\) is the inverse of \(f(x)\). The ordered pair (2,25) is a solution of the function \(g(x)\). a. Write this solution using function notation. b. Write an ordered pair that we know to be a solution of \(f(x)\) c. Use the answer to part (b) and write the solution using function notation.

Step-by-Step Solution

Answer

a. \(g(2) = 25\) b. (25, 2) c. \(f(25) = 2\)

Step 1: Understand Function Notation

For the function \(g(x)\), the notation \(g(2) = 25\) indicates that when the input \(x = 2\), the output is 25. Thus, the ordered pair used in function notation would be \(g(2) = 25\).

Step 2: Determine Ordered Pair for the Inverse

The inverse function \(f(x)\) implies that if \(g(x)\) maps \(x\) to 25, then \(f(x)\) maps 25 back to \(x = 2\). Therefore, the ordered pair for the function \(f(x)\) would be (25, 2).

Step 3: Write the Inverse Solution in Function Notation

Using the ordered pair (25, 2) for the function \(f(x)\), we can write it in function notation as \(f(25) = 2\). This shows that when the input is 25 for the function \(f(x)\), the output is 2.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. This essentially means they "undo" the action of exponential functions. When you have an exponential equation like \(a^b = c\), the logarithmic form would be \(\log_a(c) = b\). This is a very fundamental concept in mathematics since it allows for solving equations where the variable is in the exponent.

In the context of the given exercise, the function \(f(x) = \log_5 x\) is the logarithmic function. It maps an output back to the input of its inverse, the exponential function \(g(x) = 5^x\). So, if you have an output of 25 in the exponential function \(g(2) = 25\), using the logarithmic function \(f(x)\), you'd return to its input, forming the ordered pair \((25, 2)\).

Understanding that logarithmic functions "undo" the exponential functions is crucial for solving problems that involve these inverses. It helps in many real-world applications, such as compound interest calculations and solving problems in physics and engineering.

Exponential Functions

Exponential functions have the form \(g(x) = a^x\), where \(a\) is a constant and \(x\) is the variable. These functions grow at a rate proportional to their current value, making them useful in modeling growth processes in biology, finance, and many other fields.

In the given exercise, the exponential function is \(g(x) = 5^x\). This function takes any real number as input and maps it to its exponential growth. For example, with \(x = 2\), \(g(2) = 5^2 = 25\), showing the steep growth characteristic of exponential functions. This exponential function \(g(x)\) acts as the inverse to our earlier logarithmic function \(f(x)\) such that \(g(2) = 25\) pairs with \(f(25) = 2\).

Understanding exponential functions is important because they model many natural phenomena, including population growth and radioactive decay, and they are used extensively in financial calculations involving interest rates.

Function Notation

Function notation is a way to denote functions clearly using symbols. It represents the output of a function and specifies which input corresponds to which output.

In the problem, function notation is used to express the relationship between inputs and outputs of functions \(g(x)\) and \(f(x)\). For \(g(x) = 5^x\), the notation \(g(2) = 25\) means when you plug 2 into the function, the result is 25. For its inverse, the logarithm function \(f(x) = \log_5 x\), we write \(f(25) = 2\) in function notation. Here, it represents that when the input is 25, the output is 2.

This notation simplifies the representation of function evaluations and helps in keeping track of input-output pairs, making it easier to understand mathematical functions. It is a standardized way of calling out a specific point within a function's domain and codomain, facilitating easier analysis and computation of function values.