Question

PLEASE HELP!!!
Let $f(x)=x^{4}-2x^{3} -3x^{2} +4x+4$, $g(x)=\sqrt{x^{2} -x-2}$ and $h(x)=\frac{-x^{2} +1}{x^{2} -x-2}$

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)

Answer 1

The domain of a function is simply the possible x-values of the function

• The domain of g(x) is a subset of the domain of f(x)
• The range of g(x) is a subset of the range of f(x)
• The domain of f(x) is a subset of the breaks in the domain of h(x)

The domain of f(x) and g(x)

From the graphs of f(x) and g(x) (see attachment), we have the following observations

• The domain of f(x) is $\mathbf{(-\infty, \infty)}$, because the function extends in both directions of the x-axis
• The domain of g(x) is $\mathbf{(-\infty, -1]\ u\ [2, \infty)}$, because the function cannot take any value of x between -1 and 2

By comparing the domain of both functions,

The domain of g(x) is a subset of the domain of f(x)

The range of f(x) and g(x)

From the graphs of f(x) and g(x), we have the following observations

• The range of f(x) is $\mathbf{(-\infty, \infty)}$, because the function extends in both directions of the y-axis
• The range of g(x) is $\mathbf{[0,\infty)}$, because the function does not take any negative y value

By comparing the range of both functions,

The range of g(x) is a subset of the range of f(x)

The breaks in the domain of h(x) and the zeros of f(x)

From the graphs of f(x) and h(x), we have the following observations

• The zeros of f(x) are -3,778, -0.711 and 1.489; because the graph of f(x) crosses the x-axis at these points
• The domain of h(x) is $\mathbf{\:\left(-\infty \:,\:-1\right)\cup \left(-1,\:2\right)\cup \left(2,\:\infty \:\right)}$, because the graph of h(x) does not take any x value at x = 2

So, the breaks in the domain of h(x) is between -1 and 2

By comparing the zeros of f(x) and the breaks in the domain of h(x),

The domain of f(x) is a subset of the breaks in the domain of h(x)

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