How do you find the domain and the range of the function #f(x)= x^2 - 2x -3#?

Answer 1

Domain: #(-oo, oo)# Range: #[-4, oo)#

In general, the domain, or #x# values which yield an output #f(x),# of a polynomial function is all real numbers, denoted by #(-oo, oo)# in interval notation.
Range becomes more specific. This is a quadratic function. In general, the range, or #y-#values for which the function exists, of a quadratic with vertex at #(h,k)# is #[k, oo)#, OR #(-oo, k]# if the parabola is inverted (it isn't in this case -- we don't begin with #-x^2#).

Let's locate the vertex now.

We have #f(x)=ax^2+bx+c=x^2-2x-3, a=1, b=-2, c=-3#
#h=-b/(2a)=2/2=1#
#k=f(h)=f(1)=1-2-3=-4#
The range is then #[-4, oo]#
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Answer 2

To find the domain of the function (f(x) = x^2 - 2x - 3), we consider the values of (x) for which the function is defined. Since it's a polynomial function, the domain is all real numbers.

To find the range, we analyze the behavior of the function. Since (f(x) = x^2 - 2x - 3) is a quadratic function, its graph is a parabola. By examining the vertex of the parabola, we determine the minimum or maximum value. In this case, since the coefficient of (x^2) is positive, the parabola opens upwards, and its vertex represents the minimum value of the function. We can use the vertex formula (x = -\frac{b}{2a}) to find the x-coordinate of the vertex, then substitute it into the function to find the corresponding y-coordinate. This gives us the minimum value of the function, and the range will be all real numbers greater than or equal to this minimum value. Alternatively, we can use calculus to find the derivative of the function and locate its critical points to determine the minimum value and hence the range.

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Answer from HIX Tutor

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.

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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