# How do I find the range of the function #f(x)=10-x^2#?

The range of

Let us look at some details.

Since

so the largest value of

Since

Hence, the range is

The graph of

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To find the range of the function ( f(x) = 10 - x^2 ), we need to determine the set of all possible output values or y-values that the function can produce.

The given function is a quadratic function of the form ( f(x) = ax^2 + bx + c ), where ( a = -1 ), ( b = 0 ), and ( c = 10 ). Since the coefficient of the ( x^2 ) term is negative, the parabola opens downwards.

The maximum value of the function occurs at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula ( x = -\frac{b}{2a} ).

Substituting the values of ( a ) and ( b ) into the formula, we get:

[ x = -\frac{0}{2(-1)} = 0 ]

To find the corresponding y-coordinate (or the maximum value of the function), we substitute ( x = 0 ) into the function:

[ f(0) = 10 - (0)^2 = 10 ]

So, the vertex of the parabola is at the point ( (0, 10) ), and the maximum value of the function is 10.

Since the parabola opens downwards, the range of the function is all real numbers less than or equal to the maximum value. Therefore, the range of the function ( f(x) = 10 - x^2 ) is ( { y \mid y \leq 10 } ), or in interval notation, ( (-\infty, 10] ).

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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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