Functions Part 1

Given, f(x)=2x−1;f:R→R
g(x)‌‌=‌

x−1∕2

x−1

;g:R−{1}→R
f[g(x)]‌‌=2g(x)−1
‌‌=2×(‌

x− 1 2

x−1

)−1=2×(‌

2x−1

2(x−1)

)−1
‌‌=‌

2x−1

x−1

−1=‌

2x−1−x+1

x−1

=‌

x

x−1

∴‌‌f[g(x)]‌‌=1+‌

1

x−1

Now, draw the graph of 1+‌

1

x−1

,because Any horizontal line does not cut the graph at more than one points, so it is one-one and here, co-domain and range are not equal, so it is into.Hence, the required function is one-one into.