Sure! Let's go through a detailed, step-by-step solution to find the function [tex]\( f(x) \)[/tex].
The given function is [tex]\( f(x) = -2x + 3 \)[/tex].
### Step-by-Step Breakdown:
1. Identify the structure of the function:
The function [tex]\( f(x) \)[/tex] is a linear function of the form [tex]\( f(x) = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
2. Determine the slope and y-intercept:
In the function [tex]\( f(x) = -2x + 3 \)[/tex]:
- The coefficient of [tex]\( x \)[/tex] (which is [tex]\( -2 \)[/tex]) represents the slope [tex]\( m \)[/tex].
- The constant term (which is [tex]\( 3 \)[/tex]) represents the y-intercept [tex]\( b \)[/tex].
3. Understanding the slope:
The slope [tex]\( m \)[/tex] of -2 means that for every unit increase in [tex]\( x \)[/tex], the value of [tex]\( f(x) \)[/tex] will decrease by 2 units.
4. Understanding the y-intercept:
The y-intercept [tex]\( b \)[/tex] of 3 indicates that when [tex]\( x = 0 \)[/tex], the value of [tex]\( f(x) \)[/tex] will be 3.
5. Example calculations:
To better understand, let's calculate a few values:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -2(0) + 3 = 3 \][/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = -2(1) + 3 = -2 + 3 = 1 \][/tex]
- When [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = -2(-1) + 3 = 2 + 3 = 5 \][/tex]
6. Graphical representation:
Plotting these points on a graph can help visualize the function:
- (0, 3)
- (1, 1)
- (-1, 5)
Connecting these points with a straight line will give you the graph of the function [tex]\( f(x) = -2x + 3 \)[/tex].
### Summary:
The function [tex]\( f(x) = -2x + 3 \)[/tex] is a linear function with:
- A slope (rate of change) of -2
- A y-intercept (starting value) of 3
You can use this function to determine the output [tex]\( f(x) \)[/tex] for any given input [tex]\( x \)[/tex].
