To solve the function [tex]\( f(x) = 2x - 3 \)[/tex], let's take a detailed step-by-step approach.
Let's break it down as follows:
### Definition
1. The function [tex]\( f(x) \)[/tex] represents a linear relationship between [tex]\( x \)[/tex] and [tex]\( f(x) \)[/tex].
2. In this function, [tex]\( x \)[/tex] is the independent variable, and [tex]\( f(x) \)[/tex] is the dependent variable.
### Calculation Steps
1. Identify the coefficients:
- The coefficient of [tex]\( x \)[/tex] is 2.
- The constant term (y-intercept) is -3.
2. Behavior of the Function:
- This is a linear function with a slope of 2.
- The slope indicates that for every unit increase in [tex]\( x \)[/tex], [tex]\( f(x) \)[/tex] will increase by 2 units.
- The y-intercept is -3, meaning when [tex]\( x = 0 \)[/tex], [tex]\( f(x) \)[/tex] will be -3.
### Important Points on the Graph
1. Intercepts:
- Y-Intercept: When [tex]\( x = 0 \)[/tex], [tex]\( f(0) = 2(0) - 3 = -3 \)[/tex].
- X-Intercept: Set [tex]\( f(x) = 0 \)[/tex] to find where the function crosses the x-axis:
[tex]\[
0 = 2x - 3 \implies 2x = 3 \implies x = \frac{3}{2}
\][/tex]
### Example Values
To illustrate the function with a few example values:
1. When [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 2(1) - 3 = 2 - 3 = -1
\][/tex]
2. When [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = 2(-1) - 3 = -2 - 3 = -5
\][/tex]
3. When [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 2(2) - 3 = 4 - 3 = 1
\][/tex]
### Summary
- The function [tex]\( f(x) = 2x - 3 \)[/tex] is a straight line with a slope of 2.
- It has a y-intercept at (0, -3).
- The x-intercept is at [tex]\(\left(\frac{3}{2}, 0\right)\)[/tex].
By following these steps, we understand the linear function [tex]\( f(x) = 2x - 3 \)[/tex] and its characteristics.
