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Functions, 2ADV EQ-Bank 14

The function  \(y=f(x)\)  is defined by:

\begin{align*}
f(x)= \begin{cases}|x+1|-2, & \text { for }\ x \leqslant 1 \\ x^2-4, & \text { for }\ x>1\end{cases}
\end{align*}

  1. Sketch  \(y=f (x)\)   (3 marks)

  2. For what values of \(x\) is  \(f(x)=0\)?   (1 mark)

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a.   
     
 
b.    \(f(x)=0\ \ \text{when}\ \ x=-3,2\).

Show Worked Solution

a.   
     
 
b.    \(f(x)=0\ \ \text{when}\ \ x=-3,2\).

Functions, 2ADV EQ-Bank 24

The temperature \(T\) (in °C) in a greenhouse follows the pattern:

\begin{align*}
T(h)= \begin{cases}10+2 h, & \text {for }\ 0 \leqslant h<6 \\ 22, & \text {for }\ 6 \leqslant h \leqslant 18 \\ 58-2 h, & \text {for }\ 18<h \leqslant 24\end{cases}
\end{align*}

where \(h\) is the number of hours after midnight.

  1. Sketch the graph of \(T(h)\) for  \(0 \leqslant h \leqslant 24\)   (2 marks)

  2. At what time(s) during the day is the temperature exactly 18 °C?   (2 marks)

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a.
     
 

b.   \(\text{Temperature is 18° at 4 am and 8 pm.}\)

Show Worked Solution

a.
     
 

b.   \(\text{Temperature}=18^{\circ} \ \text{twice (see graph)}\)

\(10+2 h=18 \ \Rightarrow \ h=4\)

\(58-2 h=18 \ \Rightarrow \ h=20\)

\(\therefore \ \text{Temperature is 18° at 4 am and 8 pm.}\)

Functions, 2ADV EQ-Bank 25

Consider the function  \(h(x)=\begin{cases}\dfrac{x^2-9}{x-3}, & \text {for } x \neq 3 \\ k, & \text {for } x=3\end{cases}\)

  1. For what value of \(k\) is \(h(x)\) continuous at  \(x =3\)?   (2 marks)

  2. Sketch  \(y=h (x)\)  for this value of \(k\).   (2 marks) 

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a.    \(\text{Since} \ \ \dfrac{x^2-9}{x-3}=\dfrac{(x+3)(x-3)}{x-3}=x+3\)

\(\text{As} \ \ x \rightarrow 3, x+3 \rightarrow 6\)

\(h(x) \ \text {is continuous when}\ \  k=6\)
 

b.   
         

Show Worked Solution

a.    \(\text{Since} \ \ \dfrac{x^2-9}{x-3}=\dfrac{(x+3)(x-3)}{x-3}=x+3\)

\(\text{As} \ \ x \rightarrow 3, x+3 \rightarrow 6\)

\(h(x) \ \text {is continuous when}\ \  k=6\)
 

b.   
         

Functions, 2ADV EQ-Bank 16

Consider the function 

\begin{align*}
f(x)=\begin{cases}-x^2+4, & \text {for }\ x<1 \\ 2 x+1, & \text {for} \ 1 \leq x<3 \\ 7, & \text {for }\ x \geq 3\end{cases}
\end{align*}

  1. Sketch  \(y=f (x)\)   (3 marks)

  2. State the range of  \(f(x)\)   (1 mark)

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a.
       
 

b.   \(\text {Range:} \ \ y \in(-\infty, 7]\)

Show Worked Solution

a.
       
 

b.   \(\text {Range:} \ \ y \in(-\infty, 7]\)

Functions, 2ADV EQ-Bank 20

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

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

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

Functions, 2ADV EQ-Bank 17

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

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

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

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

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

b.
       

Show Worked Solution

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

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

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

b.
       

Functions, 2ADV EQ-Bank 15

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

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

  1. Sketch  \(y=f(x)\)   (3 marks)

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

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a.
   

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

Show Worked Solution

a.
   

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