To find the [tex]\( y \)[/tex]-intercept of the linear function [tex]\( f(x) = \frac{2}{9} x + \frac{1}{3} \)[/tex], we need to focus on the constant term in the equation. This constant term is what [tex]\( f(x) \)[/tex] equals when [tex]\( x = 0 \)[/tex].
Here’s the step-by-step process:
1. Understand the structure of a linear equation: A linear function is generally written in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.
2. Identify the [tex]\( y \)[/tex]-intercept: In the equation [tex]\( f(x) = \frac{2}{9} x + \frac{1}{3} \)[/tex], you need to identify the constant term, which represents the [tex]\( y \)[/tex]-intercept. This constant term is the value of the function when [tex]\( x = 0 \)[/tex].
3. Set [tex]\( x \)[/tex] to 0:
[tex]\[
f(0) = \frac{2}{9} (0) + \frac{1}{3}
\][/tex]
4. Calculate the value:
[tex]\[
f(0) = 0 + \frac{1}{3}
\][/tex]
[tex]\[
f(0) = \frac{1}{3}
\][/tex]
The [tex]\( y \)[/tex]-intercept is therefore [tex]\( \frac{1}{3} \)[/tex].
Given the options:
- [tex]\(-\frac{2}{9}\)[/tex]
- [tex]\(-\frac{1}{3}\)[/tex]
- [tex]\(\frac{1}{3}\)[/tex]
- [tex]\(\frac{7}{9}\)[/tex]
The correct answer is [tex]\( \frac{1}{3} \)[/tex].
