# Thread: Find the range of a function

1. ## Find the range of a function

How do you algebraicly find the range of a function. I can tell what the range is supposed to be by looking at the graph, but I cannot show it using algebra.

2. Originally Posted by redier
How do you algebraicly find the range of a function. I can tell what the range is supposed to be by looking at the graph, but I cannot show it using algebra.
Excellent question. It used to bother me too, until I developed a trick.
Consider a function,
$\displaystyle y=f(x)$
The range are all the values of $\displaystyle y$ such that there exists an $\displaystyle x$ in the domain such as,
$\displaystyle y=f(x)$.
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Example,
Find the range of,
$\displaystyle y=\frac{1}{1+x^2}$
The domain is all real numbers.
Thus, for what values of $\displaystyle y$ does the equation have a solution.
Say,
$\displaystyle y=\frac{1}{1+x^2}$
Then,
$\displaystyle \frac{1}{y}=1+x^2$
Thus,
$\displaystyle \frac{1}{y}-1=x^2$
Note this equation can only have a solution when the Left Hand side is non-negative. That is,
$\displaystyle \frac{1}{y}-1\geq 0$.
Thus,
$\displaystyle \frac{1}{y}\geq 1$
If you solve this inequality (details omitted) you have that,
$\displaystyle 0<y\leq 1$
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As an excersce,
Find the range of,
$\displaystyle y=\sin x+\cos x$