Question: Write the slope-intercept equation of the function $f$ whose graph satisfies the given conditions.
The graph of $f$ is perpendicular to the line whose equation is $6x - 5y - 10 = 0$ and has the same $y$-intercept as this line.
The equation of the function using function notation is $f(x) =$
1. **State the problem:** We need to find the equation of a function $f(x)$ whose graph is perpendicular to the line $6x - 5y - 10 = 0$ and shares the same $y$-intercept as this line.
2. **Find the slope and $y$-intercept of the given line:**
Rewrite the line in slope-intercept form $y = mx + b$:
$$6x - 5y - 10 = 0 \implies -5y = -6x + 10 \implies y = \frac{6}{5}x - 2$$
So, the slope $m_1 = \frac{6}{5}$ and the $y$-intercept $b = -2$.
3. **Find the slope of the perpendicular line:**
The slope of a line perpendicular to another with slope $m_1$ is the negative reciprocal:
$$m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{6}{5}} = -\frac{5}{6}$$
4. **Write the equation of the perpendicular line with the same $y$-intercept:**
Using slope-intercept form:
$$f(x) = m_2 x + b = -\frac{5}{6}x - 2$$
5. **Final answer:**
$$\boxed{f(x) = -\frac{5}{6}x - 2}$$
This is the equation of the function $f$ whose graph is perpendicular to the given line and has the same $y$-intercept.