If #f(x)=x^2# and #g(x)=-12x+7#, how do you find the domain and range of f(x), g(x), and f(g(x))?

Answer 1

Consider the function input values that work or don't work, and consider the possible function output values from all input values. Specific answers given below.

Please disregard the following sentence if you are not familiar with it; it is not relevant to your goals. We'll assume that we are dealing with real numbers here, that is, not complex numbers.

#f(x)=x^2# can be applied to any #x#, so its domain is #[-oo,oo]#. Its output is always positive - because squaring turns a number from negative to positive. So its range is #[0,oo]#.
#g(x)=-12x+7# is simpler. It takes all values, and it returns values not limited to any particular range. Both its domain and range are #[-oo,oo]#.
Happily the range of #g# matches the domain of #f#, and so the domain of #f(g(x))# is equal to the domain of #g(x)#,#[-oo,oo]# while its range is equal to the range of #f(x)#, #[0,oo]#.
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Answer 2

The domain of ( f(x) = x^2 ) is all real numbers because you can square any real number.

The range of ( f(x) = x^2 ) is all non-negative real numbers, since squaring any real number results in a non-negative value.

The domain of ( g(x) = -12x + 7 ) is all real numbers, as there are no restrictions on the input ( x ).

The range of ( g(x) = -12x + 7 ) is all real numbers because the function is a linear function, and linear functions have a range of all real numbers.

To find the domain of ( f(g(x)) ), we need to consider the values of ( x ) for which ( g(x) ) is defined. Since ( g(x) ) is defined for all real numbers, we can use any real number as input for ( g(x) ). However, we also need to consider the values of ( g(x) ) that will result in a valid input for ( f(x) = x^2 ). Since squaring any real number results in a valid output, the domain of ( f(g(x)) ) is also all real numbers.

To find the range of ( f(g(x)) ), we need to find the range of ( g(x) ) first, and then use those values as inputs for ( f(x) = x^2 ). Since the range of ( g(x) ) is all real numbers, we can use any real number as input for ( f(x) ). Therefore, the range of ( f(g(x)) ) is also all non-negative real numbers.

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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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