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Answer :
To find the values for f(x) in the function [tex]f(x) = \frac{x^2 + 2}{x - 3}[/tex], we'll evaluate it using the given options. Let's consider each option one-by-one:
Option a: [tex]x = \frac{2}{3}[/tex]
- Substitute [tex]x = \frac{2}{3}[/tex] into [tex]f(x)[/tex]:
[tex]f\left(\frac{2}{3}\right) = \frac{\left(\frac{2}{3}\right)^2 + 2}{\frac{2}{3} - 3} = \frac{\frac{4}{9} + 2}{\frac{2}{3} - 3}[/tex] - Simplify the expression:
[tex]f\left(\frac{2}{3}\right) = \frac{\frac{4}{9} + \frac{18}{9}}{\frac{2}{3} - \frac{9}{3}} = \frac{\frac{22}{9}}{\frac{-7}{3}} = \frac{22}{9} \times \frac{3}{-7} = \frac{-66}{63} = -\frac{22}{21}[/tex] - So, [tex]f\left(\frac{2}{3}\right) = -\frac{22}{21}[/tex], which is a valid number.
- Substitute [tex]x = \frac{2}{3}[/tex] into [tex]f(x)[/tex]:
Option b: [tex]x = -\frac{2}{3}[/tex]
- Substitute [tex]x = -\frac{2}{3}[/tex] into [tex]f(x)[/tex]:
[tex]f\left(-\frac{2}{3}\right) = \frac{\left(-\frac{2}{3}\right)^2 + 2}{-\frac{2}{3} - 3} = \frac{\frac{4}{9} + 2}{-\frac{2}{3} - \frac{9}{3}}[/tex] - Simplify the expression:
[tex]f\left(-\frac{2}{3}\right) = \frac{\frac{4}{9} + \frac{18}{9}}{-\frac{11}{3}} = \frac{\frac{22}{9}}{-\frac{11}{3}} = \frac{22}{9} \times \frac{-3}{11} = \frac{-66}{99} = -\frac{2}{3}[/tex] - So, [tex]f\left(-\frac{2}{3}\right) = -\frac{2}{3}[/tex].
- Substitute [tex]x = -\frac{2}{3}[/tex] into [tex]f(x)[/tex]:
Option c: undefined
- The function is undefined when the denominator is zero. In this case, [tex]f(x) = \frac{x^2 + 2}{x - 3}[/tex] is undefined for [tex]x = 3[/tex] because division by zero is not possible.
Option d: [tex]x = 0[/tex]
- Substitute [tex]x = 0[/tex] into [tex]f(x)[/tex]:
[tex]f(0) = \frac{0^2 + 2}{0 - 3} = \frac{2}{-3} = -\frac{2}{3}[/tex] - So, [tex]f(0) = -\frac{2}{3}[/tex].
- Substitute [tex]x = 0[/tex] into [tex]f(x)[/tex]:
Therefore, the function [tex]f(x)[/tex] is not undefined by the given options. If 3 were among the options, that would make the function undefined. Thus, the correct answer is none of the provided options in the context of 'undefined', and the function has outputs for the provided options a, b, and d.
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