Answer with Step-by-step explanation:
We are given that
[tex]f(x)=\frac{3x-1}{x}[/tex]
For each real number [tex]x\neq 0[/tex]
To prove that f is one -to-one.
Proof:Let [tex]x_1[/tex] and [tex]x_2[/tex] be any nonzero real numbers such that
[tex]f(x_1)=f(x_2)[/tex]
By using the definition of f to rewrite the left hand side of this equation
[tex]f(x_1)=\frac{3x_1-1}{x_1}[/tex]
Then, by using the definition of f to rewrite the right hand side of this equation of [tex]f(x_1)=f(x_2)[/tex]
[tex]f(x_2)=\frac{3x_2-1}{x_2}[/tex]
Equating the expression then we get
[tex]\frac{3x_1-1}{x_1}=\frac{3x_2-1}{x_2}[/tex]
[tex]3x_1x_2-x_2=3x_2x_1-x_1[/tex]
[tex]3x_1x_2-3x_1x_2+x_1=x_2[/tex]
[tex]x_1=x_2[/tex]
Therefore, f is one-to-one.