The domain of f(x) is all set of real values while, the domain of g(x) is x ≤ -1 or x ≥ 2 and both functions have the same range
<h3>Part A: Compare the domain and range of the function f(x) to g(x)</h3>
The functions are given as:
f(x) = x^4 - 2x^3 - 3x^2 + 4x + 4
g(x) = √(x^2 - x - 2)
<u>Domain</u>
The polynomial function f(x) has no restriction on its input.
So, the domain of f(x) is all set of real values
Set the radical of g(x) = √(x^2 - x - 2) greater than 0
x^2 - x - 2 ≥ 0
Factorize
(x + 1)(x - 2) ≥ 0
Solve for x
x ≥ -1 and x ≥ 2
Combine both inequalities
x ≤ -1 and x ≥ 2
So, the domain of g(x) is x ≤ -1 or x ≥ 2
<u>Range</u>
Using a graphical calculator, we have:
- Range of f(x) = x^4 - 2x^3 - 3x^2 + 4x + 4 ⇒ f(x) ≥ 0
- Range of g(x) = √(x^2 - x - 2) ⇒ g(x) ≥ 0
Hence, both functions have the same range
<h3>How do the breaks in the domain of h(x) relate to the zeros of f(x)?</h3>
We have:
h(x) = (-x^2 + x)/(x^2 - x - 2)
Set the denominator to 0
x^2 - x - 2 = 0
The above represents the radical of the function f(x)
This means that the breaks in the domain of h(x) and the zeros of f(x) are the same
Read more about domain and range at:
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<u>Complete question</u>
Let f(x) = x^4 - 2x^3 - 3x^2 + 4x + 4, g(x) = √(x^2 - x - 2) and h(x) = (-x^2 + x)/(x^2 - x - 2)
Part A: Use complete sentences to compare the domain and range of the polynomial function f (x) to that of the radical function g(x). (5 points)
Part B: How do the breaks in the domain of h (x) relate to the zeros of f (x)? (5 points)