 The range of a function

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Consider the real function $$f$$.

The set of all values $$f(x)$$ for $$x$$ in the domain of $$f$$ is called the range of $$f$$.

If $$p$$ is a real number in the domain of $$f$$ with $$f(p)=0$$, then $$p$$ is called a zero of $$f$$.

The range of a function depends on the domain of the function: the larger we choose the domain, the larger in general the range will be.

Let’s practice!

Consider the real function $$f(x) = \sqrt{x-9}+4$$.

The largest possible domain of $$f$$ has the form $$x\ge a$$ for a certain number $$a$$; with other words: it is of the form $$\left[a,\infty \right]$$.

The range of $$f$$ on this domain is of the form $$\left[b,\infty \right]$$ for a certain number $$b$$. Hence, it consists of all $$y$$ with $$y\ge b$$.

Determine $$a$$ and $$b$$.

The largest possible domain of $$f$$ is the set of values of $$x$$ at which $$f(x)$$ is defined. Because the argument of the root has to be greater than or equal to zero, the function $$f(x) = \sqrt{x-9}+4$$ is only defined for $$x-9\ge0$$, or $$x\ge 9$$. Therefore, the largest possible domain of $$f$$ is $$x\ge 9$$.

The range is the set of values that $$f$$ can take on this domain. The smallest value $$f$$ can take $$f(9) = 4$$. Hence, the range of $$f$$ is $$y\ge 4$$.

$$a = 9$$ and $$b = 4$$