Question

the function $y = (\frac{1}{2})^x$ is the parent function of which of the following? (1 point)
$f(x)=-(2)^x$
$f(x)=x^{\frac{1}{2}}$
$f(x)=-5(\frac{1}{2})^x$
$f(x)=-5x^2$

Explanation:

Step1: Recall parent - function concept

A parent - function is the simplest form of a function family. Transformations can be applied to it to get other functions in the family. The function $y = (\frac{1}{2})^x$ is an exponential function.

Step2: Analyze each option

• Option 1: $f(x)=-(2)^x$. The base of $y = (\frac{1}{2})^x$ is $\frac{1}{2}$, and the base of $f(x)=-(2)^x$ is 2. Also, the sign is different. They are not in the same function - family.
• Option 2: $f(x)=x^{\frac{1}{2}}=\sqrt{x}$ is a square - root function, which is a different type of function from the exponential function $y = (\frac{1}{2})^x$.
• Option 3: $f(x)=-5(\frac{1}{2})^x$ is an exponential function. It is a vertical stretch by a factor of 5 and a reflection about the $x$ - axis of the parent function $y = (\frac{1}{2})^x$. It belongs to the same function - family as $y = (\frac{1}{2})^x$.
• Option 4: $f(x)=-5x^{2}$ is a quadratic function, which is a different type of function from the exponential function $y = (\frac{1}{2})^x$.

Answer:

$f(x)=-5(\frac{1}{2})^x$