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Question
If f(x) = 2x and g(x) = `x^2/2 + 1`, then which of the following can be a discontinuous function ______.
Options
f(x) + g(x)
f(x) – g(x)
f(x) . g(x)
`("g"(x))/("f"(x))`
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Solution
If f(x) = 2x and g(x) = `x^2/2 + 1`, then which of the following can be a discontinuous function `("g"(x))/("f"(x))`.
Explanation:
We know that the algebraic polynomials are continuous functions everywhere.
∴ f(x) + g(x) is continuous .....`[(∵ "Sum, difference and product"),("of two continuous functions is"),("also continuous")]`
f(x) – g(x) is continuous
f(x) . g(x) is continuous
`("g"(x))/("f"(x))` is only continuous if g(x) ≠ 0
∴ `("f"(x))/("g"(x)) = (2x)/(x^2/2 + 1) = (4x)/(x^2 + 2)`
Here, `("g"(x))/("f"(x)) = (x^2/2 + 1)/(2x) = (x^2 + 2)/(4x)`
Which is discontinuous at x = 0.
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