Question

the tables display two functions: ( f(x) ) and ( g(x) ). function ( g(x) ) is composed of one or more transformations applied to the basic linear function ( f(x)=x ).
transforming functions
i can look at the data to determine the transformation(s) applied to ( f(x) ).

multiple transformations of linear functions
choose a transformation to change the graph.
reflection
dilation
horizontal translation
vertical translation
edit the transforming function.
( f(x) = square x )
transformation form: ( y = pm a f(x) + d )
choose a transformation.

Explanation:

To determine the transformation from \( f(x) = x \) to \( g(x) \), we analyze the input - output table:

Step 1: Analyze the relationship between input \( x \) and output \( g(x) \)

Let's take some values from the table. When \( x=-6 \), \( g(x) = 3 \); when \( x=-3 \), \( g(x)=1.5 \); when \( x = 0 \), \( g(x)=0 \); when \( x = 3 \), \( g(x)=- 1.5 \); when \( x = 6 \), \( g(x)=-3 \)

We can see that for each value of \( x \), \( g(x)=-\frac{1}{2}x \)

Step 2: Compare with the transformation form \( y=\pm A(x)+D \)

The base function is \( f(x)=x \). The general form of the transformation of a linear function is \( y=\pm A(x)+D \), where \( A \) is the dilation factor and the sign indicates reflection, and \( D \) is the vertical shift.

In our case, \( D = 0 \) (since there is no vertical shift, when \( x = 0 \), \( g(0)=0 \) which is the same as \( f(0) = 0\)). The sign is negative, which indicates a reflection over the \( x \) - axis, and \( A=\frac{1}{2} \), which indicates a dilation (compression by a factor of \( \frac{1}{2} \))

If we consider the transformation form \( y=\pm A(x)+D \), and \( f(x)=x \), then \( g(x)=- \frac{1}{2}f(x)+0 \)

So the transforming function \( g(x) \) in terms of \( f(x) \) is \( g(x)=-\frac{1}{2}f(x) \)

Answer:

The transformation applied to \( f(x)=x \) to get \( g(x) \) is a reflection over the \( x \) - axis and a dilation (compression) by a factor of \( \frac{1}{2} \). The function \( g(x)=-\frac{1}{2}x \)