For the following linear​ function, find the​ domain, range, and intercepts. Then find the minimum and maximum values of f left parenthesis x right parenthesis on the given interval. Write the domain and range in interval notation.

f left parenthesis x right parenthesis equals 5 x plus 7​; left bracket negative 3 comma 0 right bracket

Just type it the regular way, so far I can make out the following ....

f(x) = 5x + 7 ; (-3,0)

I will interpret (-3,0) as interval notation and -3 < x < 0
f(-3) = -8, f(0) = 7

domain (-3,0), range (-8,7)

the min approaches -8
the max approaches 7

words, words, words!! How about

f(x) = 5x+7 on the interval [-3,0]

Clearly the domain is [3,0]
You know this is just a straight line, so the range is all the y values between f(-3) and f(0).

Now surely you can figure the min and max values of y.

To find the domain and range of the function f(x) = 5x + 7 on the interval [-3, 0], we can follow these steps:

1. Domain:
The domain of a linear function is always the set of all real numbers. Therefore, the domain for f(x) = 5x + 7 on any interval is (-∞, ∞).

2. Range:
To find the range, we need to consider the given interval [-3, 0]. In this case, we plug in the endpoints and determine the minimum and maximum values.

When x = -3:
f(-3) = 5(-3) + 7 = -15 + 7 = -8

When x = 0:
f(0) = 5(0) + 7 = 0 + 7 = 7

So, the range of f(x) = 5x + 7 on the interval [-3, 0] is [-8, 7] in interval notation.

3. Intercepts:
To find the x-intercept, set f(x) = 0 and solve for x:
0 = 5x + 7
5x = -7
x = -7/5

Therefore, the x-intercept is (-7/5, 0).

To find the y-intercept, let x = 0 in the equation:
f(0) = 5(0) + 7
f(0) = 7

Therefore, the y-intercept is (0, 7).

4. Minimum and Maximum Values:
Since the function f(x) = 5x + 7 is a linear function, it is always increasing or decreasing. In this case, the slope is positive (+5), indicating that the line is increasing. Therefore, the minimum value occurs at the left endpoint of the interval, which is -3, and the maximum value occurs at the right endpoint, which is 0.

Minimum value: f(-3) = -8
Maximum value: f(0) = 7

To find the domain of the linear function f(x) = 5x + 7 on the interval [-3, 0], we need to determine the possible values of x within this interval.

The domain represents all the x-values for which the function is defined. In this case, there are no restrictions or exclusions on x, so the domain is simply the interval [-3, 0].

Now let's find the range of the function f(x). The range represents all the possible output values or y-values of the function.

Since f(x) = 5x + 7 is a linear function with a positive slope, the range is the set of all real numbers. In interval notation, the range can be expressed as (-∞, +∞).

Next, we'll find the x-intercept. To do this, we set f(x) equal to zero and solve for x:
5x + 7 = 0
x = -7/5

Therefore, the x-intercept is (-7/5, 0).

To find the y-intercept, we set x equal to zero and evaluate the function:
f(0) = 5(0) + 7 = 7

So the y-intercept is (0, 7).

Now let's determine the minimum and maximum values of f(x) on the given interval [-3, 0].

To find the minimum and maximum values, we need to evaluate the function at the endpoints of the interval.

At x = -3: f(-3) = 5(-3) + 7 = -8
At x = 0: f(0) = 5(0) + 7 = 7

Therefore, the minimum value of f(x) on the interval [-3, 0] is -8, and the maximum value is 7.

In summary:
- Domain: [-3, 0]
- Range: (-∞, +∞)
- X-intercept: (-7/5, 0)
- Y-intercept: (0, 7)
- Minimum value on [-3, 0]: -8
- Maximum value on [-3, 0]: 7