EXAM REVISION 2
1. The diagram shows a sketch of part of the graph $y=f(x)$ where $f(x)=3|x-4|-5$
Figure 2
a State the range of $f$.
$b$ Given that $\mathrm{f}(x)=-\frac{1}{3} x+k$, where $k$ is a constant has two distinct roots, state possible values of $k$.

Step 1 ：\(f(x) = 3|x-4| - 5\)

Step 2 ：\(f(x) = 3(x-4) - 5\) for \(x \geq 4\)

Step 3 ：\(f(x) = 3(4-x) - 5\) for \(x < 4\)

Step 4 ：\(f(4) = 3(4-4) - 5 = -5\)

Step 5 ：\(\lim_{x \to \infty} f(x) = \lim_{x \to \infty} (3(x-4) - 5) = \infty\)

Step 6 ：\(\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} (3(4-x) - 5) = \infty\)

Step 7 ：\boxed{(-\infty, -5] \cup (-5, \infty)}\)

Step 8 ：\(f(x) = -\frac{1}{3}x + k\)

Step 9 ：\(3|x-4| - 5 = -\frac{1}{3}x + k\)

Step 10 ：\(3(x-4) - 5 = -\frac{1}{3}x + k\) for \(x \geq 4\)

Step 11 ：\(3(4-x) - 5 = -\frac{1}{3}x + k\) for \(x < 4\)

Step 12 ：\(3x - 12 - 5 = -\frac{1}{3}x + k\) for \(x \geq 4\)

Step 13 ：\(12 - 3x - 5 = -\frac{1}{3}x + k\) for \(x < 4\)

Step 14 ：\(\frac{10}{3}x - 7 = k\) for \(x \geq 4\)

Step 15 ：\(\frac{8}{3}x + 7 = k\) for \(x < 4\)

Step 16 ：\(k \in \boxed{(-\infty, -7) \cup (7, \infty)}\)