Answer:
[tex]\displaystyle g(x) = \frac{1}{7}\, \left(\frac{1}{4}\right)^{x}[/tex].
Step-by-step explanation:
When the graph of a function is inverted across the [tex]y[/tex]-axis (the vertical axis), the new function value at each [tex]x[/tex] would be the same as that at [tex](-x)[/tex] in the original function. If the graph of [tex]y = f(x)[/tex] is reflected with respect to the vertical [tex]y[/tex]-axis, the expression for the new function would be [tex]y = f(-x)[/tex], same as replacing all instances of [tex]x[/tex] in the original expression with [tex](-x)[/tex].
To scale the graph of a function vertically by a constant factor of [tex]a[/tex] ([tex]a > 0[/tex] and [tex]a \ne 1[/tex],) multiply the expression for that function by this constant. For example, [tex]y = f(x)[/tex] would become [tex]y = a\, f(x)[/tex]. The function is compressed vertically if [tex]0 < a < 1[/tex], and stretched vertically if [tex]a > 1[/tex].
In this question, the scale factor would be [tex]a = (1/7)[/tex], which represents a vertical compression.
Overall, the expression for the new function obtained from these two transformations would be:
[tex]\displaystyle g(x) = \frac{1}{7}\, f(-x) = \frac{1}{7}\, \left(\frac{1}{4}\right)^{-(-x)} = \frac{1}{7}\, \left(\frac{1}{4}\right)^{x}[/tex].
