`f: R to R ` is a function where f(x)= 2x-3 . Check whether f is noe -one ?

Answer

To determine whether the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = 2x - 3 \) is one-to-one (injective), we will follow these steps:

### Step 1: Assume \( f(x_1) = f(x_2) \)
Let \( x_1 \) and \( x_2 \) be any two elements in the domain \( \mathbb{R} \). We start by assuming that the outputs of the function for these two inputs are equal:
\[ f(x_1) = f(x_2) \]

### Step 2: Substitute the function definition
Substituting the definition of the function into the equation, we have:
\[ 2x_1 - 3 = 2x_2 - 3 \]

### Step 3: Simplify the equation
Next, we can simplify this equation by adding 3 to both sides:
\[ 2x_1 = 2x_2 \]

### Step 4: Divide by 2
Now, we divide both sides by 2:
\[ x_1 = x_2 \]

### Step 5: Conclusion
Since we have shown that \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \), we conclude that the function \( f \) is one-to-one.

Thus, we can say:
\[ \text{Therefore, } f \text{ is a one-to-one function.} \]

---