Function $g$ can be thought of as a scaled version of $f(x)=|x|$. What is the equation for $g(x)$?
Answer (2)
The equation for the function g ( x ) , which is a scaled version of f ( x ) = ∣ x ∣ , is given by g ( x ) = k ∣ x ∣ , where k is a constant indicating the scale. This means that g ( x ) is obtained by multiplying f ( x ) by the constant factor k . Depending on the value of k , this scaling can stretch or compress the graph of the function.
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Recognize that a scaled version of a function involves multiplying the function by a constant.
Represent the scaled function g ( x ) as g ( x ) = k f ( x ) , where k is the scaling factor.
Substitute f ( x ) = ∣ x ∣ into the equation.
State the final equation for g ( x ) : g ( x ) = k ∣ x ∣ .
Explanation
Understanding the Problem The problem states that function g is a scaled version of f ( x ) = ∣ x ∣ . This means that g ( x ) is obtained by multiplying f ( x ) by a constant factor. Our goal is to find the equation for g ( x ) .
Setting up the Equation Let's denote the scaling factor by k . Then, the scaled function g ( x ) can be written as: g ( x ) = k ( x ) Since f ( x ) = ∣ x ∣ , we can substitute this into the equation for g ( x ) .
Finding the Equation for g(x) Substituting f ( x ) = ∣ x ∣ into the equation g ( x ) = k f ( x ) , we get: g ( x ) = k ∣ x ∣ This is the equation for g ( x ) , where k is the scaling factor.
Final Answer The equation for g ( x ) , which is a scaled version of f ( x ) = ∣ x ∣ , is: g ( x ) = k ∣ x ∣ where k is a constant.
Examples
Imagine you are adjusting the volume on a speaker. The original sound can be represented by the function f ( x ) = ∣ x ∣ , where x is the input signal. When you increase the volume, you are essentially scaling the sound. If you double the volume, the new sound can be represented by g ( x ) = 2∣ x ∣ . This scaling concept is used in audio engineering, image processing, and many other fields where adjusting the magnitude of a signal or function is necessary.
