Math  /  Algebra

QuestionI Let f(x)=2x31x+73x22f(x)=\frac{2 x^{3}-1 x+7}{3 x^{2}-2}. Find f(3)f(3) a. 23\frac{2}{3} b. 1639\frac{16}{39} c. 3 d. 1n45\frac{1 n}{45}

Studdy Solution

STEP 1

1. The function f(x)=2x31x+73x22f(x)=\frac{2 x^{3}-1 x+7}{3 x^{2}-2} is a rational function that can be evaluated at a specific point.
2. To find f(3)f(3), we need to substitute x=3x=3 into the function.

STEP 2

1. Substitute x=3x=3 into the numerator of f(x)f(x).
2. Substitute x=3x=3 into the denominator of f(x)f(x).
3. Evaluate the resulting values of the numerator and the denominator.
4. Compute f(3)f(3) by dividing the evaluated numerator by the evaluated denominator.

STEP 3

Substitute x=3x=3 into the numerator 2x3x+72x^3 - x + 7.
2(3)31(3)+7=2273+7 2(3)^3 - 1(3) + 7 = 2 \cdot 27 - 3 + 7

STEP 4

Simplify the expression in the numerator.
2273+7=543+7=58 2 \cdot 27 - 3 + 7 = 54 - 3 + 7 = 58

STEP 5

Substitute x=3x=3 into the denominator 3x223x^2 - 2.
3(3)22=392 3(3)^2 - 2 = 3 \cdot 9 - 2

STEP 6

Simplify the expression in the denominator.
392=272=25 3 \cdot 9 - 2 = 27 - 2 = 25

STEP 7

Evaluate f(3)f(3) by dividing the evaluated numerator by the evaluated denominator.
f(3)=5825 f(3) = \frac{58}{25}

STEP 8

Conclude that f(3)=5825f(3) = \frac{58}{25}. None of the provided choices match this exact fraction, so there might be a mistake in the problem statement or the provided choices. However, the correct evaluation based on the given function is:
f(3)=5825 f(3) = \frac{58}{25}

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