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Functions, 2ADV EQ-Bank 10

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 9

Consider the function

\begin{align*}
f(x)=\begin{cases}2^x, & \text {for }\ x<0 \\ m x+c, & \text {for }\ 0 \leq x \leq 2 \\ \dfrac{8}{x}, & \text {for }\ x>2\end{cases}
\end{align*}

Given that \(f(x)\) is continuous at both  \(x =0\)  and \(x =2\):

  1. Find the values of \(m\) and \(c\).   (2 marks)

  2. Identify any asymptotes of  \(y=f(x)\).   (1 mark)

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a.    \(f(x)=\left\{\begin{array}{cl}2^x & \text {for } x<0 \\ m x+c & \text {for } 0 \leq x \leq 2 \\ \dfrac{8}{x} & \text {for } x>2\end{array}\right.\)

\(\text {Continuous at}\ \ x=0:\)

\(2^\circ=m(0)+c \ \Rightarrow \ c=1\)

\(\text {Continuous at}\ \ x=2:\)

\(2 m+1=\dfrac{8}{2} \ \Rightarrow \ m=\dfrac{3}{2}\)
 

b.    \(\text{Asymptotes:}\)

\(\text{As} \ x \rightarrow-\infty, 2^x \rightarrow 0^{+}\)

\(\text{As} \ x \rightarrow \infty, \dfrac{8}{x} \rightarrow 0^{+}\)
 

\(\text{Asymptote at} \ \ y=0.\)

\(\text{There are no vertical asymptotes.}\)

Show Worked Solution

a.    \(f(x)=\left\{\begin{array}{cl}2^x & \text {for } x<0 \\ m x+c & \text {for } 0 \leq x \leq 2 \\ \frac{8}{x} & \text {for } x>2\end{array}\right.\)

\(\text {Continuous at}\ \ x=0:\)

\(2^\circ=m(0)+c \ \Rightarrow \ c=1\)

\(\text {Continuous at}\ \ x=2:\)

\(2 m+1=d\dfrac{8}{2} \ \Rightarrow \ m=\dfrac{3}{2}\)
 

b.    \(\text{Asymptotes:}\)

\(\text{As} \ x \rightarrow-\infty, 2^x \rightarrow 0^{+}\)

\(\text{As} \ x \rightarrow \infty, \dfrac{8}{x} \rightarrow 0^{+}\)
 

\(\text{Asymptote at} \ \ y=0.\)

\(\text{There are no vertical asymptotes.}\)

Functions, 2ADV EQ-Bank 8

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 7

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 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  \((0,-5)\).

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

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

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a.    \(a=4, b=-5\)

b.    \(17\)

Show Worked Solution

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

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

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

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

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

\(2 a-5=3 \ \ \Rightarrow\ \ a=4\)
 

b.     \(g(3)-g(-1)\) \(=\left(3^2-1\right)-[4(-1)-5]\)
    \(=8+9\)
    \(=17\)

Functions, 2ADV EQ-Bank 5

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 4 MC

A parking garage charges according to the following piecewise function 

\(C(t)= \begin{cases}5, & \text{for }\ 0<t \leq 1 \\ 5+3(t-1), & \text{for }\ 1<t \leq 4 \\ 18, & \text{for }\ t>4\end{cases}\) 

where \(C\) is the cost in dollars and \(t\) is time in hours.

How much does it cost to park for 3.5 hours?

  1. $11.50
  2. $12.50
  3. $13.50
  4. $18.00
Show Answers Only

\(B\)

Show Worked Solution

\(C(3.5)=5+3(3.5-1)=\$ 12.50\)

\(\Rightarrow B\)

Functions, 2ADV EQ-Bank 3 MC

For the function  \(f(x)=\left\{\begin{array}{ll}|x+2|, & \text{for } x \leq 0 \\ \sqrt{x}, & \text{for } x>0\end{array}\right.\),  what is  \(f (-2)+ f (4)\) ?

  1. \(2\)
  2. \(4\)
  3. \(6\)
  4. \(8\)
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\(A\)

Show Worked Solution

\(f(x)= \begin{cases}|x+2|, & \text { for } x \leq 0 \\ \sqrt{x}, & \text { for } x>0\end{cases}\)

\(f(-2)=|-2+2|=0\)

\(f(4)=\sqrt{4}=2\)

\(f(-2)+f(4)=2\)

\(\Rightarrow A\)

Functions, 2ADV EQ-Bank 2 MC

The function  \(f(x)=\left\{\begin{array}{ll}k x+2, & \text{for } x<-1 \\ x^2+3, & \text{for } x \geq-1\end{array}\right.\) 

If \(f(x)\) is continuous at  \(x =-1\), what is the value of \(k\) ?

  1. \(-2\)
  2. \(-1\)
  3. \(1\)
  4. \(2\)
Show Answers Only

\(A\)

Show Worked Solution

\(f(x) \ \text{ is continuous at} \ \ x=-1:\)

\(k(-1)+2=(-1)^2+3 \ \ \Rightarrow\ \ k=-2\)

\(\Rightarrow A\)

Functions, 2ADV EQ-Bank 1 MC

Which of the following piecewise functions is not continuous at  \(x =1\) ?

  1. \(f(x)= \begin{cases}x^2+1, & \text{for } \ x \leq 1 \\ 3 x-1, & \text{for } x>1\end{cases}\)
  2. \(f(x)= \begin{cases}2 x, & \text{for } \ x<1 \\ x+1, & \text{for } x \geq 1\end{cases}\)
  3. \(f(x)= \begin{cases}-x^2, & \text{for }\ x \leq 1 \\ 2 x-1, & \text{for } x>1\end{cases}\)
  4. \(f(x)= \begin{cases}3-x, & \text{for } \ x<1 \\ x^2+1, & \text{for } x \geq 1\end{cases}\)
Show Answers Only

\(C\)

Show Worked Solution

\(\text {Consider option }C:\)

\(\text{As} \ \ x \rightarrow 1^{-},-x^2 \rightarrow-1\)

\(\text{As} \ \ x \rightarrow 1^{+}, 2 x-1 \rightarrow 1\)

\(\therefore \ \text {Not continuous at} \ \ x=1\)

\(\Rightarrow C\)

Functions, 2ADV EQ-Bank 6

  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 5

  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 4

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

Functions, 2ADV F1 EQ-Bank 24 MC

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  \(x=2\) ?

  1. \(1\)
  2. \(2\)
  3. \(-1\)
  4. \(-2\)
Show Answers Only

\(C\)

Show Worked Solution

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

  \(3-(x-2)^2\) \(=mx+5\)
  \(3-(2-2)^2\) \(=2m+5\)
   \(2m\) \(=-2\)
  \(m\) \(=-1\)

 
\(\Rightarrow C\)