Property 9
Piecewise Function
The f function can be defined in different ways according to the values in its domain. Functions defined as such are called piecewise functions.
\(f(x)=\begin{cases}h(x)&x\ge a\\g(x)&x<a\end{cases}\)
\(x\ge a\Rightarrow f(x)=h(x)\)
\(x<a\Rightarrow f(x)=g(x)\)
1.
\(f(x)=\begin{cases}x-3&x>2\\3x+1&x\le2\end{cases}\)
\(\Rightarrow f(1)+f(3)=?\)
2.
\(f(x)=\begin{cases}\sqrt{x+1}&x>3\\x^2-10&x\le3\end{cases}\)
\(\Rightarrow f(8)-f(1)=?\)
3.
\(f(x)=\begin{cases}x^2-3&x<1\\\sqrt{x}+6&x\ge1\end{cases}\)
\(\Rightarrow f(4)-f(0)=?\)
4.
\(f(x)=\begin{cases}x-2&x>1\\3&x=1\\x^2+1&x<1\end{cases}\)
\(\Rightarrow f(0)+f(1)+f(2)=?\)
Property 10
Identity Function
- \(f:A\to A\) (let it be a function)
If \(\forall x\in A,f(x)=x\) then the function f is called a identity function and is denoted by I.
Constant Function
- \(f:A\to B\) be a function and \(k\in B\).
\(\forall x\in A,f(x)=k\) then the function f is called a constant function.
\(f(x)=\samefrac{ax+b}{cx+d}\)
If \(ad=bc\) then f is a constant function.
1.
\(f:\mathbb{R}\to \mathbb{R}\)
(\(f\) constant function)
\(f(x)=(a-2)x^2+(b-1)x+a\cdot b\)
\(\Rightarrow f(5)=?\)
2.
\(f:\mathbb{R}\setminus\{-a\}\to \mathbb{R}\)
(\(f\) constant function)
\(f(x)=\samefrac{2x-6}{x+a}\)
\(\Rightarrow a=?\)
3.
\(f:\mathbb{R}\setminus\left\{-\samefrac{1}{2}\right\}\to \mathbb{R}\)
(\(f\) constant function)
\(f(x)=\samefrac{ax+8}{4x+2}\)
\(\Rightarrow a=?\)
4.
\(f:\mathbb{R}\to \mathbb{R}\)
(\(f\) identity function)
\(f(x)=(a+1)x^2+(b+2)x+c-3\)
\(\Rightarrow a\cdot b\cdot c=?\)
5.
\(f:\mathbb{R}\to \mathbb{R}\)
(\(f\) identity function)
\(f(3a-5)=2a+7\)
\(\Rightarrow a=?\)