The problem is about an affine function $g$ such that $g(0) = 3$ and $g(2) = 7$. We are asked to: 1. Determine the expression of $g(x)$.

1. Problem Description

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

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

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

3. Find the number $x$ such that $g(x) = 21$.

4. Sketch the graph of the function $g$.

2. Solution Steps

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

Since $g$ is an affine function, its expression is of the form $g(x) = ax + b$. We are given that $g(0) = 3$ and $g(2) = 7$.

Using $g(0) = 3$, we have:

$g(0) = a(0) + b = 3$, so $b = 3$.

Now we have $g(x) = ax + 3$. Using $g(2) = 7$, we have:

$g(2) = a(2) + 3 = 7$

$2a = 7 - 3 = 4$

$a = \frac{4}{2} = 2$

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

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

Using the expression $g(x) = 2x + 3$, we can calculate $g(1)$ and $g(3)$:

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

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

3. Find the number $x$ such that $g(x) = 21$.

We want to find $x$ such that $g(x) = 21$. So we set $2x + 3 = 21$ and solve for $x$:

$2x + 3 = 21$

$2x = 21 - 3 = 18$

$x = \frac{18}{2} = 9$

4. Sketch the graph of the function $g$.

The graph of $g(x) = 2x + 3$ is a straight line. We already have two points: $(0, 3)$ and $(2, 7)$. We can plot these points and draw a line through them. Alternatively, we can use the point $(9, 21)$ we found in the previous step.

3. Final Answer

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

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

3. The number is

9.

4. The graph is a line through $(0, 3)$ and $(2, 7)$.