The problem states that $g$ is an affine function such that $g(0) = 3$ and $g(2) = 7$. We need to: (1) Determine the expression of $g(x)$. (2) Calculate $g(1)$ and $g(3)$. (3) Find the number $x$ for which $f(x) = 21$, noting that there seems to be a typo and this likely means $g(x) = 21$. (4) Draw the line representing the graph of the function $g$. We will not draw the line but we can indicate the equation and points needed to draw it.

1. Problem Description

The problem states that $g$ is an affine function such that $g(0) = 3$ and $g(2) = 7$. We need to:

(1) Determine the expression of $g(x)$.

(2) Calculate $g(1)$ and $g(3)$.

(3) Find the number $x$ for which $f(x) = 21$, noting that there seems to be a typo and this likely means $g(x) = 21$.

(4) Draw the line representing the graph of the function $g$. We will not draw the line but we can indicate the equation and points needed to draw it.

2. Solution Steps

(1) Determining the expression of $g(x)$:

Since $g$ is an affine function, it has the form $g(x) = ax + b$ for some constants $a$ and $b$.

We are given that $g(0) = 3$, so $a(0) + b = 3$, which implies $b = 3$.

We are also given that $g(2) = 7$, so $a(2) + b = 7$. Since $b = 3$, we have $2a + 3 = 7$.

Subtracting 3 from both sides, we get $2a = 4$. Dividing by 2, we find $a = 2$.

Therefore, the expression for $g(x)$ is $g(x) = 2x + 3$.

(2) Calculating $g(1)$ and $g(3)$:

Using the expression $g(x) = 2x + 3$, we can calculate:

$g(1) = 2(1) + 3 = 2 + 3 = 5$.

$g(3) = 2(3) + 3 = 6 + 3 = 9$.

(3) Finding the number $x$ such that $g(x) = 21$:

We are looking for $x$ such that $g(x) = 21$. Since $g(x) = 2x + 3$, we have $2x + 3 = 21$.

Subtracting 3 from both sides, we get $2x = 18$. Dividing by 2, we find $x = 9$.

(4) Tracing the line representing the graph of the function $g$:

The equation of the line is $y = g(x) = 2x + 3$. We already know two points on the line, $(0,3)$ and $(2,7)$. We calculated $g(1)=5$, so $(1,5)$ is another point. We also found that when $g(x) = 21$, $x=9$, so $(9,21)$ is a point. The line passes through all of these points.

3. Final Answer

(1) The expression for $g(x)$ is $g(x) = 2x + 3$.

(2) $g(1) = 5$ and $g(3) = 9$.

(3) The number $x$ for which $g(x) = 21$ is $x = 9$.

(4) The line representing the graph of $g$ is given by the equation $y = 2x + 3$, and passes through points such as $(0, 3)$, $(1, 5)$, $(2, 7)$ and $(9, 21)$.