The function f is given by f(x)=x^(2)+1, and the function g is given by g(x)=((x-3))/(x). Which of the following is an expression for f(g(x)) ? (A) (x^(3)-3x^(2)+x-3)/(x) (B) (x^(2)-2)/(x^(2)+1) (C) (x^(2)-6x+9)/(x^(2))+1 (D) (x^(2)-8)/(x^(2))

Question

The function  f is given by  f(x)=x^(2)+1, and the function  g is given by  g(x)=((x-3))/(x). Which of the following is an expression for  f(g(x)) ? (A)  (x^(3)-3x^(2)+x-3)/(x) (B)  (x^(2)-2)/(x^(2)+1) (C)  (x^(2)-6x+9)/(x^(2))+1 (D)  (x^(2)-8)/(x^(2))

Bytelearn · Accepted Answer

Substitute into f(x): First, we need to substitute the expression for g(x) into the function f(x). f(x) = x^2 + 1 g(x) = (x - 3) / x So, f(g(x)) = f((x - 3) / x)Replace x in f(x): Now we will replace every instance of x in f(x) with (x - 3) / x. f(g(x)) = ((x - 3) / x)^2 + 1Expand binomial square: Next, we will expand the square of the binomial ((x - 3) / x)^2. f(g(x)) = ((x - 3)^2 / x^2) + 1 f(g(x)) = ((x^2 - 6x + 9) / x^2) + 1Combine terms: Now we will combine the terms over a common denominator. f(g(x)) = (x^2 - 6x + 9) / x^2 + (x^2 / x^2) f(g(x)) = (x^2 - 6x + 9 + x^2) / x^2Simplify numerator: We simplify the numerator by combining like terms. f(g(x)) = (2x^2 - 6x + 9) / x^2Check for mistakes: We notice that the expression (2x^2 - 6x + 9) / x^2 is not among the answer choices, so we must have made a mistake. Let's go back and check our work.

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