The problem consists of five questions. 1. Simplify $3(x+5) - x(x-2)$.

1. Problem Description

The problem consists of five questions.

1. Simplify $3(x+5) - x(x-2)$.

2. Solve the equation $x^2 = 3x$.

3. Evaluate $x^{\frac{1}{2}} = 4$.

4. Factorize $2x^2 - x - 15$.

5. Given that $f(x) = \frac{2x+7}{7}$ and $g(x) = \frac{3x-6}{6}$, find a. $g(6)$ and b. $f^{-1}(x)$.

2. Solution Steps

1. Simplify $3(x+5) - x(x-2)$:

$3(x+5) - x(x-2) = 3x + 15 - x^2 + 2x = -x^2 + 5x + 15$

2. Solve the equation $x^2 = 3x$:

$x^2 - 3x = 0$

$x(x-3) = 0$

$x = 0$ or $x = 3$

3. Evaluate $x^{\frac{1}{2}} = 4$:

Square both sides:

$(x^{\frac{1}{2}})^2 = 4^2$

$x = 16$

4. Factorize $2x^2 - x - 15$:

We look for two numbers that multiply to $2 \cdot (-15) = -30$ and add up to $-1$. These numbers are $-6$ and $5$.

$2x^2 - x - 15 = 2x^2 - 6x + 5x - 15 = 2x(x-3) + 5(x-3) = (2x+5)(x-3)$

5. Given that $f(x) = \frac{2x+7}{7}$ and $g(x) = \frac{3x-6}{6}$:

a. $g(6) = \frac{3(6) - 6}{6} = \frac{18 - 6}{6} = \frac{12}{6} = 2$

b. To find $f^{-1}(x)$, we set $y = f(x) = \frac{2x+7}{7}$ and solve for $x$.

$7y = 2x + 7$

$7y - 7 = 2x$

$x = \frac{7y - 7}{2}$

Therefore, $f^{-1}(x) = \frac{7x - 7}{2}$

3. Final Answer

1. $-x^2 + 5x + 15$

2. $x = 0, 3$

3. $x = 16$

4. $(2x+5)(x-3)$

5. a. $g(6) = 2$

b. $f^{-1}(x) = \frac{7x - 7}{2}$