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Find the domain and range of the function. f(x)=(1)/(x-3)+4

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The domain and range of a function represent the set of possible input and output values, respectively.

Let's begin by finding the domain of the function `f(x) = 1/(x - 3) + 4`.

The denominator in the fraction is `(x-3)`. For this function to be defined, the denominator cannot be equal to zero, as division by zero is undefined in mathematics.

Setting `x - 3 = 0`, we find that `x = 3`. So, the value `x = 3` makes the denominator zero, which means we must exclude this value from the domain.

Therefore, the domain of the function `f(x) = 1/(x - 3) + 4` is all real numbers except for `x = 3`.

Now let's determine the range of the function.

A rational function like `f(x) = 1/(x - 3) + 4` can produce any real number as an output. The input can get arbitrarily close to the number that makes the denominator zero (in this case, `x = 3`), thus making the function value arbitrarily large in the negative or the positive direction. Also, by choosing `x` sufficiently large or sufficiently small, the term `1/(x - 3)` can get arbitrarily close to zero, so the whole function value can get as close as required to `4`.

Thus, the range of the function `f(x) = 1/(x - 3) + 4` is all real numbers.

To sum up, the domain is "all real numbers except 3", and the range is "all real numbers".

User Drodri
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