Which Function Is Equivalent To $g(x) = 7^{-x+3} + 17$?A. $g(x) = 7^{x+3} + 1$B. $g(x) = \left(\frac{1}{7}\right)^{x+3} + 1$C. $g(x) = \left(\frac{1}{9}\right)^{x-3} + 1$D. $g(x) = 7^{x-3} + 1$

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**Which Function is Equivalent to $g(x) = 7^{-x+3} + 17$?**

Understanding the Problem

In this article, we will explore the concept of equivalent functions and how to identify them. We will focus on a specific problem where we need to find the equivalent function to g(x)=7−x+3+17g(x) = 7^{-x+3} + 17. This problem requires a deep understanding of algebraic manipulations and properties of exponents.

What are Equivalent Functions?

Equivalent functions are functions that produce the same output for a given input. In other words, if we have two functions, f(x)f(x) and g(x)g(x), then they are equivalent if f(x)=g(x)f(x) = g(x) for all values of xx in their domain.

Properties of Exponents

To solve this problem, we need to understand the properties of exponents. Specifically, we need to know that:

  • a−x=1axa^{-x} = \frac{1}{a^x}
  • ax+y=ax⋅aya^{x+y} = a^x \cdot a^y
  • ax−y=axaya^{x-y} = \frac{a^x}{a^y}

Step 1: Simplify the Given Function

The given function is g(x)=7−x+3+17g(x) = 7^{-x+3} + 17. We can simplify this function by using the properties of exponents.

g(x)=7−x+3+17g(x) = 7^{-x+3} + 17 g(x)=17x−3+17g(x) = \frac{1}{7^{x-3}} + 17

Step 2: Rewrite the Function in a Different Form

We can rewrite the function in a different form by using the property a−x=1axa^{-x} = \frac{1}{a^x}.

g(x)=17x−3+17g(x) = \frac{1}{7^{x-3}} + 17 g(x)=17x−3+171g(x) = \frac{1}{7^{x-3}} + \frac{17}{1}

Step 3: Simplify the Function Further

We can simplify the function further by combining the two fractions.

g(x)=17x−3+171g(x) = \frac{1}{7^{x-3}} + \frac{17}{1} g(x)=1+17⋅7x−31g(x) = \frac{1 + 17 \cdot 7^{x-3}}{1}

Step 4: Identify the Equivalent Function

Now, we need to identify the equivalent function from the given options.

A. g(x)=7x+3+1g(x) = 7^{x+3} + 1 B. g(x)=(17)x+3+1g(x) = \left(\frac{1}{7}\right)^{x+3} + 1 C. g(x)=(19)x−3+1g(x) = \left(\frac{1}{9}\right)^{x-3} + 1 D. g(x)=7x−3+1g(x) = 7^{x-3} + 1

Answer

After simplifying the given function, we get:

g(x)=1+17⋅7x−31g(x) = \frac{1 + 17 \cdot 7^{x-3}}{1}

Comparing this with the given options, we can see that the equivalent function is:

B. g(x)=(17)x+3+1g(x) = \left(\frac{1}{7}\right)^{x+3} + 1

Q&A

Q: What is the equivalent function to g(x)=7−x+3+17g(x) = 7^{-x+3} + 17?

A: The equivalent function is g(x)=(17)x+3+1g(x) = \left(\frac{1}{7}\right)^{x+3} + 1.

Q: How do we simplify the given function?

A: We can simplify the given function by using the properties of exponents.

Q: What are the properties of exponents that we need to know?

A: We need to know the following properties of exponents:

  • a−x=1axa^{-x} = \frac{1}{a^x}
  • ax+y=ax⋅aya^{x+y} = a^x \cdot a^y
  • ax−y=axaya^{x-y} = \frac{a^x}{a^y}

Q: How do we rewrite the function in a different form?

A: We can rewrite the function in a different form by using the property a−x=1axa^{-x} = \frac{1}{a^x}.

Q: How do we simplify the function further?

A: We can simplify the function further by combining the two fractions.

Q: How do we identify the equivalent function?

A: We can identify the equivalent function by comparing the simplified function with the given options.