# If #f(x)=-2x^2-5x# and #g(x)=3x+2#, how do you find the domain and range of f(x), g(x), and f(g(x))?

See below.

We must obtain the form in order to determine the max/min value.

Simplify by converting to a binomial square:

We only need to consider the leading term and degree to determine end behavior.

Range then is:

Domain:

Range:

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To find the domain and range of ( f(x) = -2x^2 - 5x ), ( g(x) = 3x + 2 ), and ( f(g(x)) ), we examine each function separately.

- For ( f(x) = -2x^2 - 5x ):

Domain: Since ( f(x) ) is a quadratic function, it is defined for all real numbers ( x ). Range: To find the range, we analyze the behavior of the quadratic function. Since the leading coefficient is negative, the parabola opens downwards, and its vertex represents the maximum point. Therefore, the range of ( f(x) ) is all real numbers less than or equal to the y-coordinate of the vertex.

- For ( g(x) = 3x + 2 ):

Domain: ( g(x) ) is a linear function, defined for all real numbers ( x ). Range: Since ( g(x) ) is a linear function, its range is all real numbers.

- For ( f(g(x)) ):

First, we need to find the composition ( f(g(x)) ), which means we substitute ( g(x) ) into ( f(x) ):

[ f(g(x)) = -2(3x + 2)^2 - 5(3x + 2) ]

Now, we simplify this expression to find the domain and range of ( f(g(x)) ).

Domain: The domain of ( f(g(x)) ) is the set of all real numbers for which the expression ( (3x + 2) ) is defined within the original domain of ( f(x) ). Range: To find the range of ( f(g(x)) ), we analyze the behavior of the quadratic function formed by ( f(g(x)) ). We follow similar steps as before to determine its range.

Once we have simplified ( f(g(x)) ), we analyze its behavior to determine its domain and range.

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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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