To find the vertical and horizontal intercepts of the given function [tex]\( g(x) = (x-1)(x+2)(x-4) \)[/tex], we follow these steps:
### Vertical Intercept:
The vertical intercept occurs where the graph of the function intersects the y-axis. This happens where [tex]\( x = 0 \)[/tex].
1. Substitute [tex]\( x = 0 \)[/tex] into the function [tex]\( g(x) \)[/tex]:
[tex]\[
g(0) = (0-1)(0+2)(0-4)
\][/tex]
2. Simplify the expression:
[tex]\[
g(0) = (-1)(2)(-4)
\][/tex]
3. Calculate the product:
[tex]\[
g(0) = 8
\][/tex]
Hence, the vertical intercept is [tex]\( 8 \)[/tex].
### Horizontal Intercepts:
The horizontal intercepts occur where the graph of the function intersects the x-axis. This happens where [tex]\( g(x) = 0 \)[/tex].
1. Set the function equal to zero:
[tex]\[
(x-1)(x+2)(x-4) = 0
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
(x-1) = 0 \implies x = 1
\][/tex]
[tex]\[
(x+2) = 0 \implies x = -2
\][/tex]
[tex]\[
(x-4) = 0 \implies x = 4
\][/tex]
Hence, the horizontal intercepts are [tex]\( x = -2 \)[/tex], [tex]\( x = 1 \)[/tex], and [tex]\( x = 4 \)[/tex].
### Summary:
- The vertical intercept is [tex]\( 8 \)[/tex].
- The horizontal intercepts are [tex]\( -2 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( 4 \)[/tex].
So, filling in the blanks:
Given the function [tex]\( g(x) = (x-1)(x+2)(x-4) \)[/tex], its vertical intercept is [tex]\(\boxed{8}\)[/tex] and its horizontal intercepts are [tex]\(\boxed{-2, 1, 4}\)[/tex].
