Problem 90 - Algebra A Combined Function

Question

Let \(f(x)=\log _{0.3} x\). Then \(g(x)=0.3^{x}\) is the inverse of \(f(x)\). The ordered pair (3,0.027) 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. \( g(3) = 0.027 \) b. \((0.027, 3)\) c. \( f(0.027) = 3 \)

Step 1: Understand the Inverse Relationship

Given that \( f(x) = \log_{0.3} x \), the function \( g(x) = 0.3^x \) is its inverse. This means any point \((a, b)\) on \( g(x) \) corresponds to a point \((b, a)\) on \( f(x) \).

Step 2: Use Function Notation for Given Solution of g(x)

We are given the ordered pair \((3, 0.027)\) as a solution of \( g(x) = 0.3^x \). Using function notation, this is written as \( g(3) = 0.027 \).

Step 3: Find Ordered Pair for f(x)

Since \((3, 0.027)\) is a point on \( g(x) \), its inverse \( f(x) \) will have the point \((0.027, 3)\).

Step 4: Use Function Notation for Solution of f(x)

The solution found in step 3 corresponds to the function notation \( f(0.027) = 3 \) for the function \( f(x) = \log_{0.3} x \).

Key Concepts

Logarithmic Functions

Logarithmic functions are an essential concept in mathematics, acting as the inverse operations of exponential functions. They are particularly useful when dealing with exponential growth or decay, where you need to solve for the exponent. The general form of a logarithmic function is given by \( f(x) = \log_b(x) \), where \( b \) is the base of the logarithm. If the base \( b \) is less than 1, as in our example where \( b = 0.3 \), the function decreases as \( x \) increases.
Logarithms can help solve equations where the variable is an exponent. This inverse relationship means if \( b^y = x \), then \( y = \log_b(x) \). For instance, with \( f(x) = \log_{0.3}(x) \), finding \( f(0.027) = 3 \) reflects the principle that 0.3 raised to the power of 3 equals 0.027.

• Logarithms undo an exponent, finding the power to which the base must be raised to get the given number.
• They are highly useful in real-world applications like measuring sound intensity (in decibels) or the pH level in chemistry.

Exponential Functions

Exponential functions are one of the building blocks of mathematics, defined as functions where the variable appears in the exponent. The basic form of an exponential function is \( g(x) = b^x \), where \( b \) is a positive constant called the base. These functions are characterized by their rapid increase or decrease; if \( b > 1 \) the function rises, and if \( 0 < b < 1 \), like \( g(x) = 0.3^x \), the function declines.
In an exponential function, small changes in \( x \) can lead to significant changes in the output value \( g(x) \). This property is why they are often used to model growth or decay processes, such as population growth or radioactive decay.

• Exponential functions grow by a constant factor over equal intervals, showing a constant rate of percentage growth.
• The graph of an exponential function is a curved line, getting steeper or shallower depending on the base.

This behavior makes them the inverse of logarithmic functions, which helps in understanding points like the pair (3, 0.027) for \( g(x) \). This pair reflects that when 0.3 is raised to the third power, the result is 0.027.

Function Notation

Function notation is a systematic way of expressing functions, allowing us to convey the relationship between inputs and outputs clearly. It's typically written as \( f(x) \), meaning that the function \( f \) is dependent on the variable \( x \). This notation provides an easy way to plug in values and find corresponding outputs.
For example, in the function \( g(x) = 0.3^x \), if \( x = 3 \), we write \( g(3) = 0.027 \). This shows that when 3 is the input, 0.027 is the output. Similarly, for \( f(x) = \log_{0.3}(x) \), the notation \( f(0.027) = 3 \) exemplifies that when 0.027 is the input, 3 is the output.

• Function notation is a concise way to communicate which variable is dependent on which input, helping in understanding algebraic expressions.
• It allows complex relationships to be easily understandable, specifying variable roles unmistakably.

Using function notation enhances the clarity of mathematical expressions and makes it straightforward to identify solutions, like the ordered pairs found in our exercise.