Functions, 2ADV EQ-Bank 6

The function \(g(x)=\left\{\begin{array}{ll}a x+b, & \text {for } x \leq 2 \\ x^2-1, & \text {for } x>2\end{array}\right.\)

\(g(x)\) is continuous at \(x =2\) and passes through the point \((0,-5)\).

Find the values of \(a\) and \(b\) (2 marks)

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Evaluate \(g(3)-g(-1)\) (1 mark)

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a. \(a=4, b=-5\)

b. \(17\)

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a. \(\text{Since} \ f(x) \ \text{is continuous at} \ \ x=2:\)

\(a(2)+b\)	\(=2^2-1\)
\(2 a+b\)	\(=3\ \ldots\ (1)\)

\(\text{Since} \ g(x) \ \text{passes through}\ (0,-5):\)

\(a(0)+b=-5 \ \ \Rightarrow\ \ b=-5\)

\(\text{Substitute}\ \ b=-5 \ \ \text{into (1):}\)

\(2 a-5=3 \ \ \Rightarrow\ \ a=4\)