# How do you find the domain and range of #f(x) = sqrt x / (x^2 + x - 2)#?

Domain is

Range is

Asymptotes :

The asymptotes keep the two branches of the graph, in #Q_1 and

Q_4 in that order.

View the Socratic graph for illustration.

[-10, 10, -5, 5]} graph{(sqrtx/(x^2+x-2)-y)(x-.99+.01y) =0

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To find the domain of the function ( f(x) = \frac{\sqrt{x}}{x^2 + x - 2} ), we need to identify any values of ( x ) that would result in division by zero or a square root of a negative number. The denominator cannot be zero, so we solve the equation ( x^2 + x - 2 = 0 ) to find the values of ( x ) that make the denominator zero. The solutions are ( x = -2 ) and ( x = 1 ). Therefore, the domain of the function is all real numbers except ( x = -2 ) and ( x = 1 ).

To find the range of the function, we consider the behavior of the function as ( x ) approaches positive and negative infinity. As ( x ) approaches positive infinity, both the numerator and denominator of the function approach infinity, so the function approaches zero. As ( x ) approaches negative infinity, both the numerator and denominator of the function approach negative infinity, so the function approaches zero. Therefore, the range of the function is all real numbers except ( f(x) = 0 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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