Answer :
Sure, let's go through the problem step-by-step for the linear function \( f(x) = 2x - 8 \). We'll determine the x-intercept, y-intercept, domain, range, and slope.
### (a) x-intercept:
The x-intercept is found by setting the function equal to zero and solving for \( x \).
[tex]\[ f(x) = 2x - 8 \][/tex]
Set \( f(x) = 0 \):
[tex]\[ 0 = 2x - 8 \][/tex]
Solve for \( x \):
[tex]\[ 2x = 8 \][/tex]
[tex]\[ x = \frac{8}{2} \][/tex]
[tex]\[ x = 4 \][/tex]
Thus, the x-intercept is \( 4 \).
### (b) y-intercept:
The y-intercept is found by setting \( x = 0 \) and evaluating the function at that point.
[tex]\[ f(x) = 2x - 8 \][/tex]
Set \( x = 0 \):
[tex]\[ f(0) = 2(0) - 8 \][/tex]
[tex]\[ f(0) = -8 \][/tex]
Thus, the y-intercept is \( -8 \).
### (c) Domain:
For a linear function like \( f(x) = 2x - 8 \), the domain is all real numbers because there are no restrictions on the values \( x \) can take.
Thus, the domain is \( (-\infty, \infty) \).
### (d) Range:
Similarly, the range of a linear function is also all real numbers, as the function can take any value by varying \( x \).
Thus, the range is \( (-\infty, \infty) \).
### (e) Slope of the Line:
The slope of a linear function in the form \( f(x) = mx + b \) is given by the coefficient \( m \) of \( x \). In this function, \( f(x) = 2x - 8 \), the coefficient of \( x \) is 2.
Thus, the slope is \( 2 \).
### Summary:
- x-intercept: \( 4 \)
- y-intercept: \( -8 \)
- domain: \( (-\infty, \infty) \)
- range: \( (-\infty, \infty) \)
- slope: \( 2 \)
To graph this line, you'd plot the x-intercept at [tex]\( (4, 0) \)[/tex] and the y-intercept at [tex]\( (0, -8) \)[/tex], then draw a straight line through these points, extending infinitely in both directions. The slope tells you that for every unit increase in [tex]\( x \)[/tex], [tex]\( f(x) \)[/tex] increases by 2 units.
### (a) x-intercept:
The x-intercept is found by setting the function equal to zero and solving for \( x \).
[tex]\[ f(x) = 2x - 8 \][/tex]
Set \( f(x) = 0 \):
[tex]\[ 0 = 2x - 8 \][/tex]
Solve for \( x \):
[tex]\[ 2x = 8 \][/tex]
[tex]\[ x = \frac{8}{2} \][/tex]
[tex]\[ x = 4 \][/tex]
Thus, the x-intercept is \( 4 \).
### (b) y-intercept:
The y-intercept is found by setting \( x = 0 \) and evaluating the function at that point.
[tex]\[ f(x) = 2x - 8 \][/tex]
Set \( x = 0 \):
[tex]\[ f(0) = 2(0) - 8 \][/tex]
[tex]\[ f(0) = -8 \][/tex]
Thus, the y-intercept is \( -8 \).
### (c) Domain:
For a linear function like \( f(x) = 2x - 8 \), the domain is all real numbers because there are no restrictions on the values \( x \) can take.
Thus, the domain is \( (-\infty, \infty) \).
### (d) Range:
Similarly, the range of a linear function is also all real numbers, as the function can take any value by varying \( x \).
Thus, the range is \( (-\infty, \infty) \).
### (e) Slope of the Line:
The slope of a linear function in the form \( f(x) = mx + b \) is given by the coefficient \( m \) of \( x \). In this function, \( f(x) = 2x - 8 \), the coefficient of \( x \) is 2.
Thus, the slope is \( 2 \).
### Summary:
- x-intercept: \( 4 \)
- y-intercept: \( -8 \)
- domain: \( (-\infty, \infty) \)
- range: \( (-\infty, \infty) \)
- slope: \( 2 \)
To graph this line, you'd plot the x-intercept at [tex]\( (4, 0) \)[/tex] and the y-intercept at [tex]\( (0, -8) \)[/tex], then draw a straight line through these points, extending infinitely in both directions. The slope tells you that for every unit increase in [tex]\( x \)[/tex], [tex]\( f(x) \)[/tex] increases by 2 units.
