Answer :
To explore the range of the function \( f(x) = 2 \left(\frac{1}{4}\right)^x \) after it has been reflected over the \( y \)-axis, let's go through a detailed analysis step by step.
1. Understand the original function:
The original function is \( f(x) = 2 \left(\frac{1}{4}\right)^x \).
2. Reflect the function over the \( y \)-axis:
Reflecting a function \( f(x) \) over the \( y \)-axis means we replace \( x \) with \( -x \). Thus, the reflected function is:
[tex]\[ f(-x) = 2 \left(\frac{1}{4}\right)^{-x} \][/tex]
3. Simplify the reflected function:
Recall that \( \left(\frac{1}{4}\right)^{-x} = 4^x \). Therefore, the reflected function simplifies to:
[tex]\[ f(-x) = 2 \cdot 4^x \][/tex]
4. Analyze the range of the simplified reflected function:
- The expression \( 4^x \) for any real number \( x \) is always positive.
- Since \( 4^x \) is always positive, multiplying it by 2 (a positive constant) will still result in a positive value.
- Therefore, \( 2 \cdot 4^x \) is always greater than 0 for any real number \( x \).
5. Determine the range of the reflected function:
As discussed, \( f(-x) = 2 \cdot 4^x \) results in values that are always positive. Thus, the range of the function \( f(x) = 2 \left(\frac{1}{4}\right)^x \) after it has been reflected over the \( y \)-axis is all real numbers greater than 0.
Hence, the best description of the range of the reflected function is:
All real numbers greater than 0.
1. Understand the original function:
The original function is \( f(x) = 2 \left(\frac{1}{4}\right)^x \).
2. Reflect the function over the \( y \)-axis:
Reflecting a function \( f(x) \) over the \( y \)-axis means we replace \( x \) with \( -x \). Thus, the reflected function is:
[tex]\[ f(-x) = 2 \left(\frac{1}{4}\right)^{-x} \][/tex]
3. Simplify the reflected function:
Recall that \( \left(\frac{1}{4}\right)^{-x} = 4^x \). Therefore, the reflected function simplifies to:
[tex]\[ f(-x) = 2 \cdot 4^x \][/tex]
4. Analyze the range of the simplified reflected function:
- The expression \( 4^x \) for any real number \( x \) is always positive.
- Since \( 4^x \) is always positive, multiplying it by 2 (a positive constant) will still result in a positive value.
- Therefore, \( 2 \cdot 4^x \) is always greater than 0 for any real number \( x \).
5. Determine the range of the reflected function:
As discussed, \( f(-x) = 2 \cdot 4^x \) results in values that are always positive. Thus, the range of the function \( f(x) = 2 \left(\frac{1}{4}\right)^x \) after it has been reflected over the \( y \)-axis is all real numbers greater than 0.
Hence, the best description of the range of the reflected function is:
All real numbers greater than 0.
