Question

consider the piecewise function shown below.

f(x)=\begin{cases} a|x - 12|+14, & 0leq xlt21\\ -4sqrt{x - 21}+26, & 21leq xleq30 end{cases}

in order for ( f(x) ) to be continuous over its entire domain ( (0leq xleq30) ), what must be the numerical value of parameter ( a ) in the absolute value part of this piecewise function?

Explanation:

Step1: Find the value of the second - part function at \(x = 21\)

For the function \(y=-4\sqrt{x - 21}+26\) with \(x = 21\), substitute \(x=21\) into the function:
\(y=-4\sqrt{21 - 21}+26\)
\(y = 26\)

Step2: Find the value of the first - part function at \(x = 21\)

For the function \(y=a|x - 12|+14\) with \(x = 21\), substitute \(x = 21\) into the function: \(y=a|21 - 12|+14=a\times9 + 14=9a+14\)

Step3: Set the two values equal for continuity

Since the function \(f(x)\) is continuous at \(x = 21\), we set \(9a+14=26\)
Subtract 14 from both sides: \(9a=26 - 14\)
\(9a=12\)
Divide both sides by 9: \(a=\frac{12}{9}=\frac{4}{3}\)

Answer:

\(\frac{4}{3}\)