Question

section 4.1 - inverse functions
question 6 of 10 (1 point) | question attempt: 1 of unlimited
part: 0 / 4
part 1 of 4
(a) show that ( f ( x ) = 2 x - 1 ) defines a one - to - one function.
a function is one - to - one if it can be shown that if ( f ( a ) = f ( b ) ), then ( square = square ). assume ( f ( a ) = f ( b ) ).

Explanation:

Step1: Recall the definition of one - to - one function

A function \(y = f(x)\) is one - to - one if for every \(a\) and \(b\) in the domain of \(f\), \(f(a)=f(b)\) implies \(a = b\).

Step2: Analyze the given function \(f(x)=2x - 1\)

We start with the assumption \(f(a)=f(b)\). Substitute \(x=a\) and \(x = b\) into the function \(f(x)=2x-1\). So we have \(2a - 1=2b - 1\).

Step3: Solve the equation \(2a - 1=2b - 1\)

Add \(1\) to both sides of the equation: \(2a-1 + 1=2b - 1+1\), which simplifies to \(2a=2b\).

Step4: Divide both sides by \(2\)

Dividing \(2a = 2b\) by \(2\) gives \(a = b\). So, by the definition of a one - to - one function, \(f(x)=2x - 1\) is one - to - one.

Answer:

\(a\); \(b\)