Property 17
Example
\(f:\mathbb{Z}\to \mathbb{Z}\)
\(f(x+1)=f(x)+7\)
\(f(1)=5\)
\(\Rightarrow f(10)=?\)
Solution
\(\begin{array}{l}\text{(for) }x=1\quad\cancel{f(2)}=f(1)+7\\\text{(for) }x=2\quad\cancel{f(3)}=\cancel{f(2)}+7\\\text{(for) }x=3\quad\cancel{f(4)}=\cancel{f(3)}+7\\\text{(for) }x=4\quad\cancel{f(5)}=\cancel{f(4)}+7\\\vdots\qquad\cancel{\vdots}\qquad\cancel{\vdots}\\\text{(for) }x=9\quad f(10)=\cancel{f(9)}+7\\\hline f(10)=f(1)+7\cdot9\end{array}\)
\(\hspace{85px}\downarrow\ \text{(9 terms)}\)
\(f(10)=5+63=68\)
1.
\(f:\mathbb{Z}\to \mathbb{Z}\)
\(f(x+1)=f(x)+3\)
\(f(1)=2\)
\(\Rightarrow f(4)=?\)
2.
\(f:\mathbb{Z}\to \mathbb{Z}\)
\(f(x+1)-f(x)=x\)
\(f(2)=3\)
\(\Rightarrow f(4)=?\)
3.
\(\samefrac{f(x+1)}{f(x)}=4\)
\(f(1)=2\)
\(\Rightarrow f(4)=?\)
4.
\(f:\mathbb{Z}\to \mathbb{Z}\)
\(f(x+1)=f(x)+x+3\)
\(f(2)=5\)
\(\Rightarrow f(20)=?\)
5.
\(f:\mathbb{Z}\to \mathbb{Z}\)
\(f(x)=2\cdot f(x-1)\)
\(f(5)=4\)
\(\Rightarrow f(30)=?\)
Property 18
Example
\(f(x)=2^{x-1}\)
What is the value of \(f(x+3)\) in terms of \(f(x)\)?
Solution
- \(f(x+3)=2^{(x+3)-1}=2^{x+2}\cdots\enclose{circle}{1}\)
- \(f(x)=2^{x-1}=\samefrac{2^x}{2}\Rightarrow2^x=2f(x)\cdots\enclose{circle}{2}\)
\(\text{From }\enclose{circle}{1}\text{ and }\enclose{circle}{2}\)
\(f(x+3)=2^{x+2}=2^x\cdot2^2=2\cdot f(x)\cdot2^2=8f(x)\)
1.
\(f(x)=4^{x+1}\)
What is the value of \(f(x+1)\) in terms of \(f(x)\)?
2.
\(f(x)=3^{x+2}\)
What is the value of \(f(x+2)\) in terms of \(f(x)\)?
3.
\(f(x-2)=5^x\)
What is the value of \(f(x+1)\) in terms of \(f(x)\)?
4.
\(f(x)=4x\)
What is the value of \(f(x+3)\) in terms of \(f(x)\)?
5.
\(f(x)=3x+1\)
What is the value of \(f(x+1)\) in terms of \(f(x)\)?
6.
\(f(x)=\samefrac{x+1}{x}\)
What is the value of \(f(2x)\) in terms of \(f(x)\)?