Finding the Domain and Range of a Function

Find the domain of the following function ${\log }_e\frac{2+x}{2-x}$.

Find the domain of the following function ${\sin }^{-1}\left({\log }_2\frac{x^2}{2}\right)$.

Find the domain of the following function $f\left(x\right)=\frac{1}{\sqrt{\left(1-x\right)\left(x-2\right)}}$.

Show that the function $\frac{x^2+3x+5}{x^2-5x+4}$ is undefined at $x=1 \mathrm{and} x=4$, and $\frac{\sin x}{x}$ is undefined at $x=0$.

Find the values of $\theta$ for which the function $f\left(\theta \right)=\frac{\sin \theta }{\cos \theta -\sin \theta }$ is not defined.

Find the domain of the function $f\left(x\right)=\sqrt{x-1}+\sqrt{6-x}$.

If $f\left(x\right)=\sqrt{15+x-2x^2}$, find the domain of definition of $f\left(x\right)$. If possible, find the values of $f\left(0\right), f\left(4\right)$ and $f\left(-2\right)$.

Find the range of $g\left(x\right)=\sin x+\cos x$.

Find the range of values of $x$ for the following function.

$f\left(x\right)=\sqrt{x^2-2x-3}$

Find the range of values of $x$ for the following function.

$f\left(x\right)=\sqrt{8+2x-3x^2}$

Find the range of values of $x$ for the following function.

$f\left(x\right)=\sqrt{x^2-4}$

Find the domain of the definition of the following function.

Find the domain of the definition of the following function.

Find the domain of the definition of the following function.

$f\left(x\right)=x^2$

A function $f$ is completely defined by the following table. Find the domain of definition of $f$. What is its range ?

A function $f$ is completely defined by the following table. Find the domain of definition of $f$. What is its range ?

The arrow diagram shown alongside represents part of the graph of $y$, a linear function of $x$. Determine this function. A further arrow is to be drawn at $x=3$. At what point should this arrow terminate ? If the function $f$ is such that $f : Z\to Z$, where $Z$ is a set of integers, determine the range of $f$ if the domain of $f$ is from $-4$ to $3$.

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Find the range of the function $f\left(x\right)=\frac{x}{1+x^2}, x\in R$ where $R$ is the set of real numbers,

If $f:R\to R$ is a function defined by $f\left(x\right)=\frac{x^2}{1+x^2}, x\in R$ where $R$ is the set of real numbers, find the range of $x$.

The domain of the function $f\left(x\right)={\sin }^{-1}\left[{\log }_4\left(\frac{x}{4}\right)\right]+\sqrt{17x-x^2-16}$ is