Function [tex]f[/tex] is a rational function with a [tex]y[/tex]-intercept at [tex](0,2)[/tex]. If [tex]g(x)=4 f(x)[/tex], which point represents the [tex]y[/tex]-intercept of function [tex]g[/tex]?

A. [tex](0,8)[/tex]
B. [tex](0,4)[/tex]
C. [tex](0,6)[/tex]
D. [tex](0,2)[/tex]



Answer :

To determine the [tex]\( y \)[/tex]-intercept of the function [tex]\( g(x) = 4 f(x) \)[/tex], we need to analyze what happens to the [tex]\( y \)[/tex]-intercept of the original function [tex]\( f(x) \)[/tex] when it is scaled by a factor of 4.

1. Identify the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex]:
Given that the [tex]\( y \)[/tex]-intercept of [tex]\( f(x) \)[/tex] is at [tex]\( (0, 2) \)[/tex], it means that when [tex]\( x = 0 \)[/tex] in the function [tex]\( f(x) \)[/tex], the value of [tex]\( f(x) \)[/tex] is 2. Mathematically, this is written as:
[tex]\[ f(0) = 2 \][/tex]

2. Determine the effect of scaling on [tex]\( f(x) \)[/tex]:
The function [tex]\( g(x) \)[/tex] is defined as [tex]\( g(x) = 4 f(x) \)[/tex]. To find the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex], we need to evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 4 f(0) \][/tex]

3. Substitute the value of [tex]\( f(0) \)[/tex]:
We know from step 1 that [tex]\( f(0) = 2 \)[/tex]. Substituting this into the expression for [tex]\( g(0) \)[/tex], we get:
[tex]\[ g(0) = 4 \times 2 = 8 \][/tex]

Therefore, the [tex]\( y \)[/tex]-intercept of [tex]\( g(x) \)[/tex] is at the point [tex]\((0, 8)\)[/tex].

So, the point that represents the [tex]\( y \)[/tex]-intercept of the function [tex]\( g \)[/tex] is:
[tex]\[ (0, 8) \][/tex]

Thus, the correct answer is:
[tex]\[ (0, 8) \][/tex]