Answer :
Let's analyze the linear function \( f(x) = 2x - 8 \) step-by-step:
### (a) x-intercept
To find the x-intercept, we need to set \( f(x) = 0 \) and solve for \( x \).
[tex]\[ 0 = 2x - 8 \][/tex]
Solving for \( x \):
[tex]\[ 2x = 8 \implies x = 4 \][/tex]
Thus, the x-intercept is \( (4, 0) \).
### (b) y-intercept
To find the y-intercept, we need to set \( x = 0 \) and solve for \( f(x) \).
[tex]\[ y = 2(0) - 8 \][/tex]
Simplifying this:
[tex]\[ y = -8 \][/tex]
Thus, the y-intercept is \( (0, -8) \).
### (c) Domain
The domain of any linear function is all real numbers. This is because a line extends infinitely in both directions along the x-axis.
So, the domain is \( \text{all real numbers} \).
### (d) Range
Similarly, the range of any linear function is all real numbers. This is because a line extends infinitely in both directions along the y-axis.
So, the range is \( \text{all real numbers} \).
### (e) Slope of the line
The slope of the linear function \( f(x) = 2x - 8 \) is the coefficient of \( x \). Here, the slope is \( 2 \).
### Graphing the function
To graph the function \( f(x) = 2x - 8 \), we use the intercepts as reference points. Plot the x-intercept \( (4, 0) \) and the y-intercept \( (0, -8) \) on a coordinate plane. Draw a straight line through these points to represent the linear function.
In summary:
- The x-intercept is \((4, 0)\),
- The y-intercept is \((0, -8)\),
- The domain is all real numbers,
- The range is all real numbers,
- The slope is \(2\).
You can now use these points and information to graph the linear function accurately.
### (a) x-intercept
To find the x-intercept, we need to set \( f(x) = 0 \) and solve for \( x \).
[tex]\[ 0 = 2x - 8 \][/tex]
Solving for \( x \):
[tex]\[ 2x = 8 \implies x = 4 \][/tex]
Thus, the x-intercept is \( (4, 0) \).
### (b) y-intercept
To find the y-intercept, we need to set \( x = 0 \) and solve for \( f(x) \).
[tex]\[ y = 2(0) - 8 \][/tex]
Simplifying this:
[tex]\[ y = -8 \][/tex]
Thus, the y-intercept is \( (0, -8) \).
### (c) Domain
The domain of any linear function is all real numbers. This is because a line extends infinitely in both directions along the x-axis.
So, the domain is \( \text{all real numbers} \).
### (d) Range
Similarly, the range of any linear function is all real numbers. This is because a line extends infinitely in both directions along the y-axis.
So, the range is \( \text{all real numbers} \).
### (e) Slope of the line
The slope of the linear function \( f(x) = 2x - 8 \) is the coefficient of \( x \). Here, the slope is \( 2 \).
### Graphing the function
To graph the function \( f(x) = 2x - 8 \), we use the intercepts as reference points. Plot the x-intercept \( (4, 0) \) and the y-intercept \( (0, -8) \) on a coordinate plane. Draw a straight line through these points to represent the linear function.
In summary:
- The x-intercept is \((4, 0)\),
- The y-intercept is \((0, -8)\),
- The domain is all real numbers,
- The range is all real numbers,
- The slope is \(2\).
You can now use these points and information to graph the linear function accurately.
