Problem 75 - Precalculus with Limits

Question

Think About It In Exercises \(75-78\) , use properties of exponents to determine which functions (if any) are the same. $$\begin{array}{l}{f(x)=3^{x-2}} \\ {g(x)=3^{x}-9} \\\ {h(x)=\frac{1}{9}\left(3^{x}\right)}\end{array}$$

Step-by-Step Solution

Answer

Based on the properties of exponents, only the functions \(f(x)\) and \(h(x)\) are equivalent.

Step 1: Understanding Base Equality

Since all three functions are expressed in terms of powers of 3, it's possible for them to be equivalent. In an equivalent scenerio, altering the exponents or coefficients should allow us to transform one function into another.

Step 2: Compare f(x) and g(x)

A direct comparison of \(f(x) = 3^{x-2}\) and \(g(x) = 3^{x}-9\) shows they are not equivalent, because subtraction operations in the exponent field of \(f(x)\) and the multiplication field of \(g(x)\) result in different behaviors for different values of x.

Step 3: Compare f(x) and h(x)

We transform \(h(x) = \frac{1}{9}\left(3^{x}\right)\) into \(h(x) = 3^{x-2}\). Now, it's clear that \(f(x) = h(x)\). The operation in the exponent field of \(f(x)\) and the multiplication field of \(h(x)\) due to properties of exponents results in equivalent behaviors.

Step 4: Compare g(x) and h(x)

A similar approach shows \(g(x) = 3^{x}-9\) cannot be transformed to align with \(h(x) = \frac{1}{9}\left(3^{x}\right)\) as addition/subtraction operations cause different behaviors for different values of x.

Key Concepts

Function Equivalence

Function equivalence is a core concept in algebra, especially when dealing with expressions and equations. Two functions are equivalent if they produce the same output for every input. This can involve recognizing transformations such as shifts, reflections, or scalings.

To determine equivalence:

• Check if transformations on one function can yield the other.
• Consider properties of exponents and operations applied to the function.
• Recognize that equivalence implies identical results for all inputs.

In the exercise, for instance, comparing functions like \(f(x) = 3^{x-2}\) and \(h(x) = \frac{1}{9}(3^x)\) involves rewriting one to see if it matches the other. Understanding function equivalence is essential in solving for unknowns and simplifying expressions in algebra.

Exponential Functions

Exponential functions are defined by a constant base raised to a variable exponent, generally written as \(f(x) = a^{x}\). These functions are powerful in modeling growth or decay, as they depict rapid changes in value over varying inputs.

Key properties include:

• A constant base with a real-number exponent.
• Rapid growth or decay - for base greater than 1, the function grows; for 0 < base < 1, it decays.
• Always results in positive outputs if the base is positive.

In our given exercise, every function is expressed in terms of powers of 3. Recognizing this allows us to check equivalency by transforming each function to potentially match one another, using the properties of exponential functions in the manipulations.

Algebraic Transformation

Algebraic transformations are procedures used to manipulate an expression or equation into a different form, often to simplify or reveal equivalence. Common transformations include factoring, expanding, or using exponent rules.

When it comes to exponential functions:

• Rewriting exponents using properties like \(a^{m-n} = \frac{a^m}{a^n}\).
• Factoring, where possible, to match expressions.
• Multiplicative inverses to simplify expressions.

In the step-by-step solution, transforming \(h(x) = \frac{1}{9}(3^x)\) into \(3^{x-2}\) via algebraic manipulation shows equivalence with \(f(x)\). Here, \(\frac{1}{9}\) was rewritten as \(3^{-2}\), demonstrating how intricate understanding of algebraic rules aids in function transformation and equivalence checking.