Questions: Let f be the function f(x)= 3 x^(-1) for x<-1 a x+b for -1 ≤ x ≤ 1/2 4 x^(-1) for x>1/2 Find the values of a and b that make the function continuous. (Use symbolic notation and fractions where needed.)

Let f be the function f(x)= 3 x^(-1) for x<-1 a x+b for -1 ≤ x ≤ 1/2 4 x^(-1) for x>1/2 Find the values of a and b that make the function continuous. (Use symbolic notation and fractions where needed.)
Transcript text: Let $f$ be the function \[ f(x)=\left\{\begin{array}{ll} 3 x^{-1} & \text { for } x<-1 \\ a x+b & \text { for }-1 \leq x \leq \frac{1}{2} \\ 4 x^{-1} & \text { for } x>\frac{1}{2} \end{array}\right. \] Find the values of $a$ and $b$ that make the function continuous. (Use symbolic notation and fractions where needed.)
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Solution

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Solution Steps

Step 1: Solve the system of equations

Given: \[ \begin{aligned} a - b &= 3 \\ -a + b &= 8.0 \end{aligned} \] Adding the two equations together, we get: \[ 2b = 11.0 \] Solving for \(b\): \[ b = \frac{11.0}{2} = 5.5 \]

Final Answer

\[ \boxed{b = 5.5} \]

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