Find the Domain and Range f(x)=x^4-8x^3+17x^2-8x+16

Problem

ƒ(x)=x4−8*x3+17*x2−8*x+16

Solution

1. Identify the type of function. Since ƒ(x) is a polynomial, it is defined for all real numbers.

2. Determine the domain. For any polynomial, the domain is (−∞,∞)

3. Factor the expression to find the global minimum. Notice that the expression can be rewritten as a square of a quadratic: ƒ(x)=(x2−4*x+4)2

4. Simplify the quadratic inside the square. Since x2−4*x+4=(x−2)2 the function becomes ƒ(x)=((x−2)2)2=(x−2)4

5. Analyze the range. The expression (x−2)4 is always non-negative because it is raised to an even power.

6. Find the minimum value. The minimum occurs when x−2=0 which gives ƒ(2)=0 Since the leading coefficient is positive and the degree is even, the function goes to ∞ as x goes to ±∞

Final Answer

Domain: *(−∞,∞), Range: *[0,∞)

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