The problem gives the function $g(x) = x^2 - 2x + p$, where $p$ is a real number. We are also given that the point $(1, 2)$ lies on the graph of this function. We need to find the value of $p$.

1. Problem Description

The problem gives the function $g(x) = x^2 - 2x + p$, where $p$ is a real number. We are also given that the point $(1, 2)$ lies on the graph of this function. We need to find the value of $p$.

2. Solution Steps

Since the point $(1, 2)$ lies on the graph of the function $g(x)$, we know that $g(1) = 2$.

Substitute $x = 1$ into the function:

$g(1) = (1)^2 - 2(1) + p$

$g(1) = 1 - 2 + p$

$g(1) = -1 + p$

We are given that $g(1) = 2$, so we can set up the equation:

$2 = -1 + p$

Now, we solve for $p$:

$p = 2 + 1$

$p = 3$

3. Final Answer

The value of $p$ is 3.