The functions f(x) = -4x + 11 and f(x) = 2/3x - 8 are having the range of {y ∈ R | -∞ < y < ∞}.
What are the domain and range of a function?
- The domain is the set of all the possible values of x that are taken as inputs for the function.
- The range is the set of all the values(output) that are obtained for the domain of x.
- So, domain = {input values(x)} and range = {output values(y)}
Calculation:
Step 1: Finding the range for the function f(x) = -(x + 1)² - 4
The given function is quadratic in the form of a(x - h)² + k,
So, the range of the function is y ≤ k if a < 0 or y ≥ k if a > 0
For the given function, a = -1 so, a < 0. thus, the range of the given function is y ≤ k, where k = -4
∴ Range of f(x) = -(x + 1)² - 4 is: (-∞ -4], {y | y ≤ -4}
Step 2: Finding the range for the function f(x) = -4x + 11
This is a linear function. So, the range is R.
∴ Range of f(x) = -4x + 11: (-∞, ∞), { y | y ∈ R}
Step 3: Finding the range for the function f(x) = 2/3x - 8
This is also a linear function. So, the range is R.
∴ Range of f(x) = 2/3x - 8: (-∞, ∞), { y | y ∈ R}
Step 4: Finding the range for the function f(x) = 2^x + 3
This is an exponential function. So, the range is y > k
Here k = 3.
∴ Range of f(x) = 2^x+3: (3, ∞), {y | y > 3}
Step 5: Finding the range for the function f(x) = x² + 7x - 9
This is a quadratic function.
we can write it as,
f(x) = x² + 2 × x × (7/2) + (7/2)² - (7/2)² - 9
= (x + 7/2)² - 85/4
This is in the form of a(x - h)² - k. where a = 1 > 0 and k = -85/4
Since a > 0, the range of the function is y ≥ -85/4
∴ Range of f(x) = x² + 7x - 9: [-85/4, ∞), {y | y ≥ -85/4}
Therefore, the functions f(x) = -4x + 11 and f(x) = 2/3x - 8 are having the given range {y ∈ R, -∞ < y < ∞}
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