To graph the linear function [tex]\( f(x) = -5x + 3 \)[/tex], let's first understand the properties and behaviors of linear functions.
### Graphing the Function
1. Identify the slope and y-intercept:
- The function is in the form [tex]\( f(x) = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Here, [tex]\( m = -5 \)[/tex] and [tex]\( b = 3 \)[/tex].
2. Plot the y-intercept (0, 3):
- Start by plotting the point where the line crosses the y-axis, which is at (0, 3).
3. Use the slope to find another point:
- The slope [tex]\( -5 \)[/tex] means for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 5 units.
- From the point (0, 3):
- Move 1 unit to the right (to x = 1).
- Move 5 units down (to y = -2).
- This gives the point (1, -2). Plot this point.
4. Draw the line:
- Connect the two points with a straight line extending in both directions. This is the graph of the function [tex]\( f(x) = -5x + 3 \)[/tex].
### Domain and Range
- Domain:
- The domain of a linear function is all real numbers because there are no restrictions on the values [tex]\( x \)[/tex] can take.
[tex]\[
\text{Domain: } \{ x \, | \, x \in \mathbb{R} \}
\][/tex]
In simpler terms:
[tex]\[
\text{Domain: } \text{all real numbers}
\][/tex]
- Range:
- Similarly, the range of a linear function is also all real numbers. As [tex]\( x \)[/tex] takes any real value, [tex]\( y \)[/tex] (or [tex]\( f(x) \)[/tex]) can also take any real value.
[tex]\[
\text{Range: } \{ y \, | \, y \in \mathbb{R} \}
\][/tex]
In simpler terms:
[tex]\[
\text{Range: } \text{all real numbers}
\][/tex]
### Conclusion
For the linear function [tex]\( f(x) = -5x + 3 \)[/tex]:
- The domain is all real numbers.
- The range is all real numbers.
So the final answer is:
[tex]\[
\text{Domain: } \text{all real numbers}
\][/tex]
[tex]\[
\text{Range: } \text{all real numbers}
\][/tex]
