To show that [tex]\( g(n) = m \)[/tex] given that [tex]\( g(m) = n \)[/tex], let's follow these steps in detail.
1. Define the function [tex]\( g \)[/tex]:
The function [tex]\( g \)[/tex] is given by:
[tex]\[
g(x) = \frac{7}{x}
\][/tex]
2. Use the given condition [tex]\( g(m) = n \)[/tex]:
According to the problem, [tex]\( g(m) = n \)[/tex]. This means:
[tex]\[
n = \frac{7}{m}
\][/tex]
3. Substitute [tex]\( n \)[/tex] into [tex]\( g \)[/tex]:
We need to find [tex]\( g(n) \)[/tex]. By definition of the function [tex]\( g \)[/tex], we have:
[tex]\[
g(n) = \frac{7}{n}
\][/tex]
4. Express [tex]\( g(n) \)[/tex] in terms of [tex]\( m \)[/tex]:
Since [tex]\( n = \frac{7}{m} \)[/tex], substitute this value into the expression for [tex]\( g(n) \)[/tex]:
[tex]\[
g(n) = \frac{7}{\frac{7}{m}}
\][/tex]
5. Simplify the expression:
Simplifying the expression inside the fraction, we get:
[tex]\[
g(n) = \frac{7}{} / \frac{7}{m} = 7 \times \frac{m}{7}
\][/tex]
6. Finish the simplification:
[tex]\[
g(n) = m
\][/tex]
Therefore, we have shown that if [tex]\( g(m) = n \)[/tex], then [tex]\( g(n) = m \)[/tex].
