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 \(b = 4\)

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