The problem is divided into two exercises. Exercise 1 deals with a linear function $f$ with a coefficient of $\frac{3}{2}$. We need to find the expression of $f(x)$, calculate the image of 6 by $f$, calculate $f(-8)$, $f(-4)$, $f(\frac{4}{3})$, and $f(6\sqrt{5})$, and finally, find the number whose image is 21 by $f$. Exercise 2 deals with a linear function $g$ defined by $g(x) = \frac{7}{2}x$. We need to complete a table of values.

1. Problem Description

The problem is divided into two exercises.

Exercise 1 deals with a linear function $f$ with a coefficient of $\frac{3}{2}$. We need to find the expression of $f(x)$, calculate the image of 6 by $f$, calculate $f(-8)$, $f(-4)$, $f(\frac{4}{3})$, and $f(6\sqrt{5})$, and finally, find the number whose image is 21 by $f$.

Exercise 2 deals with a linear function $g$ defined by $g(x) = \frac{7}{2}x$. We need to complete a table of values.

2. Solution Steps

Exercise 1:

(1) The expression of $f(x)$ is given by $f(x) = \frac{3}{2}x$.

(2) To calculate the image of 6 by $f$, we evaluate $f(6)$:

$f(6) = \frac{3}{2}(6) = 3(3) = 9$

(3) To calculate $f(-8)$, $f(-4)$, $f(\frac{4}{3})$, and $f(6\sqrt{5})$, we substitute these values into the expression of $f(x)$:

$f(-8) = \frac{3}{2}(-8) = 3(-4) = -12$

$f(-4) = \frac{3}{2}(-4) = 3(-2) = -6$

$f(\frac{4}{3}) = \frac{3}{2}(\frac{4}{3}) = \frac{4}{2} = 2$

$f(6\sqrt{5}) = \frac{3}{2}(6\sqrt{5}) = 3(3\sqrt{5}) = 9\sqrt{5}$

(4) To find the number $x$ such that $f(x) = 21$, we solve the equation $\frac{3}{2}x = 21$:

$x = 21 \cdot \frac{2}{3} = 7 \cdot 2 = 14$

Exercise 2:

The function is defined by $g(x) = \frac{7}{2}x$.

We are given a table with $x$ values of 0, -2, 4, and an unknown value. We are also given that $g(x)$ is 21 for an unknown $x$ and -31.5 for another unknown x.

$g(0) = \frac{7}{2}(0) = 0$

$g(-2) = \frac{7}{2}(-2) = -7$

$g(4) = \frac{7}{2}(4) = 7(2) = 14$

For $g(x) = 21$, we solve $\frac{7}{2}x = 21$:

$x = 21 \cdot \frac{2}{7} = 3 \cdot 2 = 6$

For $g(x) = -31.5$, we solve $\frac{7}{2}x = -31.5$:

$x = -31.5 \cdot \frac{2}{7} = -\frac{63}{2} \cdot \frac{2}{7} = -9$

So the complete table is:

x | 0 | -2 | 6 | 4 | -9

---|---|---|---|---|---

g(x) | 0 | -7 | 21 | 14 | -31.5

3. Final Answer

Exercise 1:

(1) $f(x) = \frac{3}{2}x$

(2) $f(6) = 9$

(3) $f(-8) = -12$, $f(-4) = -6$, $f(\frac{4}{3}) = 2$, $f(6\sqrt{5}) = 9\sqrt{5}$

(4) $x = 14$

Exercise 2:

x | 0 | -2 | 6 | 4 | -9

---|---|---|---|---|---

g(x) | 0 | -7 | 21 | 14 | -31.5