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If
G
(
x
)
G(x) below represents the transformation of the graph of the main function
y
=
(
1
2
)
x
y=(
2
1

)
x
, what is the equation of
G
(
x
)
G(x)?"

To find the equation of \( G(x) \), which is a transformation of the function \( y = \left( \frac{1}{2} \right)^x \), let's examine the transformation involved.

### Step 1: Analyze the Function \( y = \left( \frac{1}{2} \right)^x \)
The base function \( y = \left( \frac{1}{2} \right)^x \) represents an exponential decay function. Its general shape starts high on the left (for \( x < 0 \)) and approaches zero as \( x \to \infty \).

### Step 2: Analyze the Transformation
From the graph, it appears that the function has been reflected over the \( y \)-axis, changing the growth direction. A reflection over the \( y \)-axis can be applied by changing \( x \) to \( -x \) in the equation.

### Step 3: Write the Transformed Equation
If we reflect \( y = \left( \frac{1}{2} \right)^x \) over the \( y \)-axis, the new equation becomes:
\[
G(x) = \left( \frac{1}{2} \right)^{-x}
\]

### Step 4: Simplify the Equation
Using the property \( a^{-x} = \frac{1}{a^x} \), we can rewrite this as:
\[
G(x) = 2^x
\]

### Final Answer
The equation of \( G(x) \) is:
\[
G(x) = 2^x
\]

Would you like more details on transformations or the properties of exponential functions?

### Further Questions
1. What would the equation be if the graph was shifted up or down?
2. How would a horizontal shift affect the equation?
3. What is the impact of reflecting the graph over the \( x \)-axis instead of the \( y \)-axis?
4. How would a vertical stretch or compression affect the equation?
5. Can you identify the domain and range of \( G(x) = 2^x \)?

### Tip
When transforming functions, remember that changes inside the function argument \( x \) (like \( -x \)) affect horizontal properties, while changes outside the function affect vertical properties.

If G(x) below represents the transformation of the graph of the main function y = (1/2)^x, what is the equation of G(x)?

Explore how to find the equation of G(x) as a transformation of y = (1/2)^x. This problem covers exponential functions and transformations, including reflections over the y-axis. Suitable for students in Grades 9-12.

Math Problem Statement

If G ( x ) G(x) below represents the transformation of the graph of the main function y

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