Discussion
A function \(f : [-7, 6) \Rightarrow R\) is defined as follows
\[ f(x) = \begin{cases} x^{2}+2x+1 & -7 \le x \lt -5 \\ x+5 & -5 \le x \le 2 \\ x-1 & 2 \lt x \lt 6 \end{cases} \]
what is the value of \(\frac{4f(-3) - 2f(4)}{f(-6) - 4f(1)}\)?
Answer: C
The value is 2.
Pick the right piece of the function for each argument:
\[ \begin{aligned} f(-3) &= -3 + 5 = 2 \\ f(4) &= 4 - 1 = 3 \\ f(-6) &= 36 - 12 + 1 = 25 \\ f(1) &= 1 + 5 = 6 \end{aligned} \]
\[ \frac{4(2) - 2(3)}{25 - 4(6)} = \frac{8 - 6}{25 - 24} = 2 \]
\(\frac{1}{2}\) is the answer turned upside down, and the other values come from using the wrong piece of \(f\) for one of the arguments, for example \(x + 5\) for \(f(-6)\).
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