f(x) =9(4)^x; The function f is defined by the given equation. If g(x) = f(x + 2), which of the.... - Free Printable

We are given:

- $ f(x) = 9(4)^x $
- $ g(x) = f(x + 2) $

We are to find which of the following equations defines $ g(x) $.

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Step 1: Understand what $ g(x) = f(x+2) $ means

This is a horizontal shift of the function $ f(x) $. Specifically, replacing $ x $ with $ x+2 $ in $ f(x) $ shifts the graph left by 2 units.

So,
$$
g(x) = f(x+2) = 9(4)^{x+2}
$$

Now simplify this expression.

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Step 2: Simplify $ 9(4)^{x+2} $

Use exponent rules:
$$
4^{x+2} = 4^x \cdot 4^2 = 4^x \cdot 16
$$

So:
$$
g(x) = 9 \cdot 4^{x+2} = 9 \cdot 16 \cdot 4^x = 144 \cdot 4^x
$$

Therefore:
$$
g(x) = 144(4)^x
$$

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Step 3: Match with answer choices

A. $ g(x) = 18(4)^x $ → ✘
B. $ g(x) = 144(4)^x $ → ✔
C. $ g(x) = 18(8)^x $ → ✘
D. $ g(x) = 81(16)^x $ → ✘

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Final Answer:

B. $ g(x) = 144(4)^x $

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Additional Check (as shown in your notes):

You wrote:
- $ f(0) = 9(4)^0 = 9 $
- $ g(0) = f(0+2) = f(2) $

Compute $ f(2) $:
$$
f(2) = 9(4)^2 = 9 \cdot 16 = 144
$$

So $ g(0) = 144 $

Now test each option at $ x = 0 $:

- A: $ 18(4)^0 = 18 $ → ✘
- B: $ 144(4)^0 = 144 $ → ✔
- C: $ 18(8)^0 = 18 $ → ✘
- D: $ 81(16)^0 = 81 $ → ✘

Only B gives $ g(0) = 144 $, so it's correct.

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✔ Answer: B. $ g(x) = 144(4)^x $