How to create a function with a
Doamin [-3,8]
Range [-2,4]
Increase (-3,1) (5,8)
Decrease (1,5)
To create a function with a domain of [-3, 8] and a range of [-2, 4], as well as specific intervals of increase and decrease, you can follow these steps:
Step 1: Determine the slope between the two increasing points.
- The interval of increase is (-3, 1) and (5, 8).
- To find the slope, use the formula: slope = (change in y) / (change in x).
- In this case, the slope would be: (8 - 1) / (5 - (-3)) = 7 / 8.
Step 2: Determine the slope between the two decreasing points.
- The interval of decrease is (1, 5).
- Use the same formula as above to find the slope: (5 - 8) / (1 - 5) = -3 / -4 = 3 / 4. Notice that the signs of the numerator and denominator have been switched.
Step 3: Write the equation of a line for each interval.
- For the interval of increase, you can use the point-slope form: y - y₁ = m(x - x₁), where (x₁, y₁) is one of the given points and m is the slope.
- Using the point (5, 8) as an example, the equation would be: y - 8 = (7/8)(x - 5).
- Simplifying, you get: y = (7/8)x + 27/8.
- For the interval of decrease, you can use the same equation but with the opposite slope.
- Using the point (1, 5) as an example, the equation would be: y - 5 = -(3/4)(x - 1).
- Simplifying, you get: y = -(3/4)x + 23/4.
Step 4: Combine the equations.
- To include the entire domain, you need to define the function for each sub-interval within the domain.
- For x ≤ 1, the equation would be: y = -(3/4)x + 23/4.
- For 1 < x < 5, the equation would be: y = (7/8)x + 27/8.
- For x ≥ 5, the equation would be: y = (7/8)x + 27/8.
Thus, the resulting function would be:
- For x ≤ 1: y = -(3/4)x + 23/4
- For 1 < x < 5: y = (7/8)x + 27/8
- For x ≥ 5: y = (7/8)x + 27/8
Make sure to check if the function satisfies the given domain and range.