Question

Find the range of the function f(x)=%20%7D%7B2x%20-%201%7D%20" id="TexFormula1" title=" \frac{ {x}^{2} + 3x + 5 }{2x - 1} " alt=" \frac{ {x}^{2} + 3x + 5 }{2x - 1} " align="absmiddle" class="latex-formula">


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Answer 1

Rewrite the numerator as

x ² + 3x + 5 = (x - 1/2)² + 4 (x - 1/2) + 27/4

Then

(x ² + 3x + 5) / (2x - 1) = 1/2 × (x ² + 3x + 5) / (x - 1/2)

… = 1/2 × ((x - 1/2)² + 4 (x - 1/2) + 27/4) / (x - 1/2)

… = 1/2 × ((x - 1/2) + 4 + 27 / (4 (x - 1/2)))

… = 1/2 x + 7/4 + 27 / (8 (x - 1/2))

which clearly has a non-removable singularity at x = 1/2, which is to say this function has a domain including including all real numbers except 1/2.

For every number other than x = 1/2, the function takes on every possible real numbers, since 1/2 x + 7/4 alone takes on all real numbers.

So:

domain = {x ∈ ℝ | x ≠ 1/2}

range = {x ∈ ℝ}

Answer 2

Photo math is very helpful with those type of problems