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Evaluate f(x) = (x5+2i)/(x3) at x=i. Options: A. i, B. -4i, C. -3, D. 2

Mathematics

ScanSolve AI Answer

Step-by-step solution

7 steps

  1. Substitute x=ix=i into the given function f(x)f(x).

    The function is f(x)=x5+2ix3f(x) = \frac{x^5+2i}{x^3}.
    Substitute x=ix=i:

    f(i)=i5+2ii3f(i) = \frac{i^5+2i}{i^3}

  2. Evaluate the powers of ii.

    Recall the properties of the imaginary unit ii:
    i1=ii^1 = i
    i2=−1i^2 = -1
    i3=i2⋅i=−1⋅i=−ii^3 = i^2 \cdot i = -1 \cdot i = -i
    i4=(i2)2=(−1)2=1i^4 = (i^2)^2 = (-1)^2 = 1
    i5=i4⋅i=1⋅i=ii^5 = i^4 \cdot i = 1 \cdot i = i

  3. Substitute these values back into the expression for f(i)f(i).

    f(i)=i+2i−if(i) = \frac{i+2i}{-i}

  4. Simplify the numerator.

    i+2i=3ii+2i = 3i

  5. Substitute the simplified numerator back into the expression.

    f(i)=3i−if(i) = \frac{3i}{-i}

  6. Simplify the fraction.

    f(i)=−3f(i) = -3

  7. Compare the result with the given options.

    A. ii
    B. −4i-4i
    C. −3-3
    D. 2

    The calculated value is −3-3, which corresponds to option C.

    The final answer is C. −3\boxed{\text{C. } -3}.

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