If $f(x)=x^2$, which equation represents function $g$?
A. $g(x)=\frac{1}{3} f(x)$
B. $g(x)=3 f(x)$
C. $g(x)=f(\frac{1}{3} x)$
D. $g(x)=f(3 x)$
Answer (1)
Substitute f ( x ) = x 2 into each option.
A. g ( x ) = 3 1 f ( x ) = 3 1 x 2
B. g ( x ) = 3 f ( x ) = 3 x 2
C. g ( x ) = f ( 3 1 x ) = ( 3 1 x ) 2 = 9 1 x 2
D. g ( x ) = f ( 3 x ) = ( 3 x ) 2 = 9 x 2 Without additional information, any of these could represent g ( x ) .
Explanation
Understanding the Problem We are given the function f ( x ) = x 2 and asked to determine which of the given options represents a function g ( x ) . The options involve transformations of the function f ( x ) . We will substitute f ( x ) = x 2 into each option to see what the resulting expression for g ( x ) is.
Evaluating the Options Let's examine each option:
A. g ( x ) = 3 1 f ( x ) = 3 1 x 2
B. g ( x ) = 3 f ( x ) = 3 x 2
C. g ( x ) = f ( 3 1 x ) = ( 3 1 x ) 2 = 9 1 x 2
D. g ( x ) = f ( 3 x ) = ( 3 x ) 2 = 9 x 2
Conclusion Since the problem does not provide any additional information or constraints to determine a specific function g ( x ) , each of the options A, B, C, and D represents a possible function g ( x ) based on the given transformations of f ( x ) . Without further information, we cannot definitively choose one option over the others.
Examples
Understanding function transformations is crucial in many real-world applications. For instance, in physics, scaling a function can represent changes in the intensity of a wave. In economics, it can model changes in production or cost functions. In computer graphics, transformations like these are used to manipulate images and objects. For example, if f ( x ) represents the brightness of an image at pixel x , then g ( x ) = 2 f ( x ) would represent an image that is twice as bright.
