Determining the image of a function

Question

I was given a function that says:

What is the image of the function $F: \Bbb Z \times \Bbb N \rightarrow \Bbb R$ given by $f(a,b) = \frac{(a-4)}{7b}$

I need help really understanding how to find an image. I did a few questions where it said to make the question = b, but that was dealing with only one variable and this question has a and b.

score 0 · Accepted Answer · answered Aug 15 '15 at 23:44

0

The image of your function is the rational numbers, $\Bbb Q$.

Clearly, if $a$ is an integer and $b$ is a natural number (which here does not include zero), then $\frac{(a-4)}{7b}$ is a rational number. So the image is included in the rational numbers.

Given any rational number $\frac pq$ with $p$ an integer, $q$ a natural number, and $\gcd(p,q)=1$, we can set $a=7p+4$ and $b=q$. We then get

$$f(a,b)=\frac{a-4}{7b}=\frac{7p+4-4}{7q}=\frac pq$$

So the rational numbers are included in the image. We conclude overall that the image is equal to the set of rational numbers.

I hope this gives you an idea on some approaches to finding an image. The general question takes creativity. Try some examples, and work at it. Practice will make you better.

answered Aug 15 '15 at 23:44

Rory Daulton

32,288

Thank you very much for the help – ChristopherW Aug 15 '15 at 23:48
why is if a is an integer and b a natural number then (a-4),7b automatically a rational number – ChristopherW Aug 16 '15 at 00:05
@ChristopherW: Because $a-4$ is also an integer, and $7b$ is also a non-zero natural number. Any integer divided by a non-zero integer yields a rational number: in fact, that is a common definition of rational number. – Rory Daulton Aug 16 '15 at 00:24

Determining the image of a function

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