Piecewise Functions (Adv-2027)

Functions, 2ADV EQ-Bank 6

Identify where the graph \(f(x)=\dfrac{\abs{x}}{x}\) is not continuous. (1 mark)

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Sketch the graph of \(f(x)\). (2 marks)

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a. \(\text {Denominator} \neq 0\)

\(f(x)\ \text{is not continuous when} \ \ x=0\)

b. \(\text{If} \ \ x>0 \ \Rightarrow \ f(x)=\dfrac{x}{x}=1\)

\(\text{If} \ \ x<0 \ \Rightarrow \ f(x)=-\dfrac{x}{x}=-1\)

Show Worked Solution

a. \(\text {Denominator} \neq 0\)

\(f(x)\ \text{is not continuous when} \ \ x=0\)

b. \(\text{If} \ \ x>0 \ \Rightarrow \ f(x)=\dfrac{x}{x}=1\)

\(\text{If} \ \ x<0 \ \Rightarrow \ f(x)=-\dfrac{x}{x}=-1\)

Identify where the graph \(f(x)=\dfrac{x^2-1}{x-1}\) is not continuous. (1 mark)

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Sketch the graph of \(f(x)\). (2 marks)

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a. \(f(x)=\dfrac{x^2-1}{x-1}=\dfrac{(x+1)(x-1)}{(x-1)}=x+1\)

\(\text{Since denominator} \neq 0\)

\(f(x) \ \ \text{is not continuous when} \ \ x=1.\)

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a. \(f(x)=\dfrac{x^2-1}{x-1}=\dfrac{(x+1)(x-1)}{(x-1)}=x+1\)

\(\text{Since denominator} \neq 0\)

\(f(x) \ \ \text{is not continuous when} \ \ x=1.\)

Consider the function \(y=f(x)\) where

\(f(x)= \begin{cases}x^2+6, & \text { for } x \leqslant 0 \\ 6, & \text { for } 0<x \leqslant 3 \\ 2^x, & \text { for } x>3\end{cases}\)

Sketch \(y=f(x)\) (3 marks)

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For what value of \(x\) is \(y=f(x)\) NOT continuous? (1 mark)

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b. \(f(x)\ \ \text {is NOT continuous at}\ \ x=3.\)

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b. \(f(x)\ \ \text {is NOT continuous at}\ \ x=3.\)

Graph the function \(y=f(x)\) where:

\(f(x)= \begin{cases}x^2, & \text { for } x \leq-1 \\ x-1, & \text { for }-1<x \leq 1 \\ -x^3, & \text { for } x>1 \end{cases}\). (3 marks)

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Given that \(f(x)=\left\{\begin{array}{ll}3-(x-2)^2, & \text { for } x \leqslant 2 \\ m x+5, & \text { for } x>2\end{array}\right.\)

What is the value of \(m\) for which \(f(x)\) is continuous at \(x=2\) ?

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\(C\)

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\(\text {If continuous at}\ x=2:\)

\(\Rightarrow C\)