Mapping/determining domain/range

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In summary, the range for $g(z) = z^2$ for $z$ in the first quadrant is $Im w > 0$. The methodical approach to determine this is by setting $z = \rho\ e^{i\ \theta}$ and using the formula $\displaystyle g(\rho, \theta) = \rho^{2}\ e^{2\ i\ \theta} = \rho^{2}\ (\cos 2\ \theta + i\ \sin 2\ \theta )$. Since $0\le\theta\le\frac{\pi}{2}$, we can deduce that $\sin 2\theta\ge0$, which means the imaginary part of $z^
  • #1
nacho-man
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find the range for
$g(z) = z^2$ for $z$ in the first quadrant, ie $Re z > 0 $ and $Im z > 0$

Why is the answer $Im w > 0$.

Similarly, how do i go about finding the range for:

$p(z) = -2z^3$ for $z$ in the quarter disk $|z|<1$, $0<Arg z<\frac{\pi}{2}$

I am confused as to how to determine the answer, what is the methodical approach to tackle this problem?
thanks.
 
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  • #2
nacho said:
find the range for
$g(z) = z^2$ for $z$ in the first quadrant, ie $Re z > 0 $ and $Im z > 0$

Why is the answer $Im w > 0$.

Setting $\displaystyle z = \rho\ e^{i\ \theta}$ is...

$\displaystyle g(\rho, \theta) = \rho^{2}\ e^{2\ i\ \theta} = \rho^{2}\ (\cos 2\ \theta + i\ \sin 2\ \theta ) (1)$

... and because is $\displaystyle 0 \le \theta \le \frac{\pi}{2}$ is...

$\displaystyle \sin 2 \theta \ge 0 \implies \text{Im}\ (z^{2}) \ge 0$...

Kind regards

$\chi$ $\sigma$
 

FAQ: Mapping/determining domain/range

What is the domain in a mapping or function?

The domain in a mapping or function refers to the set of all possible input values for the function. It is the set of values that can be plugged into the function to produce an output.

How do you determine the domain of a function?

To determine the domain of a function, you first need to look at the restrictions or limitations on the input values. This could include restrictions due to the function's definition or restrictions based on the type of function (e.g. square root or logarithmic). Once you have identified any restrictions, you can list all the possible input values that satisfy those restrictions to determine the domain.

What is the range in a mapping or function?

The range in a mapping or function refers to the set of all possible output values for the function. It is the set of values that are produced when the input values are plugged into the function.

How do you determine the range of a function?

To determine the range of a function, you need to look at the output values that are produced when all possible input values are plugged into the function. The range will be the set of all those output values. It is important to note that some functions may have restrictions on the output values as well, which would affect the range.

Why is it important to map/determine the domain and range of a function?

Mapping/determining the domain and range of a function is important because it helps us understand the behavior and limitations of the function. It also allows us to identify any potential errors or undefined values that may occur when using the function. Additionally, knowing the domain and range can help us make predictions and analyze the function's overall performance.

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