Question

what is the formula of ( g ) in terms of ( f )?
choose 1 answer:
a ( fleft(-\frac{1}{3}x
ight) )
b ( f(-3x) )
c ( fleft(\frac{1}{3}x
ight) )
d ( f(3x) )

Explanation:

Step1: Analyze horizontal compression/stretch

For a function \(y = f(kx)\), if \(|k|> 1\), the graph of \(y = f(x)\) is horizontally compressed by a factor of \(\frac{1}{|k|}\). If \(0<|k|<1\), the graph of \(y = f(x)\) is horizontally stretched by a factor of \(\frac{1}{|k|}\). Also, if \(k<0\), there is a reflection about the \(y -\)axis.
Assume the original function \(y = f(x)\) and the transformed function \(y = g(x)\).
Let's consider a point \((x_0,y_0)\) on \(y = f(x)\) such that \(y_0=f(x_0)\). For the transformed function, if we want to get the same \(y -\)value \(y_0\) for \(y = g(x)\), assume \(g(x)=f(kx)\).
Suppose the key - point (for example, the vertex or a point of interest) of \(y = f(x)\) is at \(x = a\) and the corresponding key - point of \(y = g(x)\) is at \(x=\frac{a}{3}\) (from the graph, assuming a horizontal transformation).
We know that if \(g(x)=f(kx)\), and we substitute \(x\) values. Let \(kx_0=x_1\). If \(x_0\) is the \(x -\)coordinate of a point on \(f(x)\) and \(x_1\) is the \(x -\)coordinate of the corresponding point on \(g(x)\).
If we assume a horizontal compression (since the graph of \(g(x)\) is “narrower” in the horizontal direction compared to \(f(x)\) in the non - reflected case). Let's use the property of function transformation.
If we consider the general form of horizontal transformation \(y = f(kx)\). Let's take a sample non - zero \(x\) value. Suppose \(f(x)\) has a point \((x_0,f(x_0))\) and \(g(x)\) has a point \((x_1,f(x_0))\) such that \(x_1=\frac{1}{3}x_0\) (if there is no reflection) or \(x_1 =-\frac{1}{3}x_0\) (if there is a reflection). But if we rewrite \(x_0=- 3x_1\) (assuming reflection), then \(g(x_1)=f(-3x_1)\)

Step2: Conclusion

By the rules of function transformations (horizontal compression/stretch and reflection), when we compare the graph of \(y = g(x)\) with \(y = f(x)\), using the fact that for a function \(y = f(x)\) and its transformation \(y = g(x)=f(kx)\), and analyzing the horizontal position of key - points (assuming a reflection and compression).
If we assume that the transformation from \(y = f(x)\) to \(y = g(x)\) involves a horizontal compression by a factor of \(\frac{1}{3}\) and a reflection about the \(y -\)axis. Using the formula \(y = f(kx)\) where \(k=-3\) (because \(y = f(-3x)\) implies \(|k| = 3>1\) (compression by \(\frac{1}{3}\)) and \(k=-3\) (reflection about the \(y -\)axis)

Answer:

B. \(f(-3x)\)