Define a function f on a set of real numbers as follows: f(x) = 3x − 1 x , for each real number x ≠ 0 Prove that f is one-to-one. Proof: Let x1 and x2 be any nonzero real numbers such that f(x1) = f(x2). Use the definition of f to rewrite the left-hand side of this equation. (Enter the answer as an expression in x1.) Then use the definition of f to rewrite the right-hand side of f(x1) = f(x2). (Enter the answer as an expression in x2.) Equate the expressions obtained for the left- and right-hand sides of f(x1) = f(x2), and simplify the result completely. The result is the following equation. x1 = Therefore, f is .

Respuesta :

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.