Answer:
[tex]\textsf{a)}\quad g(x) = \dfrac{1}{3} x[/tex]
[tex]\textsf{b)}\quad g(x) =16x^2[/tex]
[tex]\textsf{c)}\quad g(x) =-\sqrt{x}[/tex]
[tex]\textsf{d)}\quad g(x) = |-x| \implies g(x)=|x|[/tex]
Step-by-step explanation:
Part a)
When a function f(x) is vertically compressed by a factor of k, the new function g(x) is given by g(x) = k · f(x). This means that the y-values of the original function are scaled down by the factor k, making the graph appear "squished" towards the x-axis.
In this case, if the graph of f(x) = x is vertically compressed by a factor of 1/3, the new function g(x) is:
[tex]g(x) = \dfrac{1}{3} x[/tex]
[tex]\dotfill[/tex]
Part b)
When a function f(x) is horizontally compressed by a factor of k, the new function g(x) is given by g(x) = f(kx). If k > 1, the graph will be compressed by 1/k. If 0 < k < 1, the graph will be stretched by 1/k.
In this case, the graph of f(x) = x² is horizontally compressed by a factor of 1/4, so k = 4.
Therefore, the new function g(x) is:
[tex]g(x) = f(4x) \\\\g(x)= (4x)^2 \\\\g(x)= 16x^2[/tex]
[tex]\dotfill[/tex]
Part c)
When a function f(x) is reflected in the x-axis, the new function g(x) is given by g(x) = -f(x). This means that the y-values of the original function are negated, resulting in a reflection that flips the graph upside down across the x-axis.
In this case, if f(x) = √x is reflected in the x-axis, the new function g(x) is:
[tex]g(x) = -f(x) \\\\ g(x)=-\sqrt{x}[/tex]
[tex]\dotfill[/tex]
Part d)
When a function f(x) is reflected in the y-axis, the new function g(x) is given by g(x) = f(-x). This operation negates the x-values of the original function f(x), effectively flipping the graph across the y-axis.
In this case, if f(x) = |x| is reflected in the y-axis, then the new function g(x) is:
Therefore, the new function g(x) is:
[tex]g(x) = f(-x) \\\\ g(x) = |-x| \\\\ g(x) = |x|[/tex]
Note that for f(x) = |x|, the reflection in the y-axis does not change the function since the absolute value of -x is the same as the absolute value of x.
