Name: Algebraic Operations: Functions
Uploaded: 2019-07-19
Duration: 7 min 10 s
Description: Now that we know what functions are, what can we do with them? Can we do operations with them? Yes! What won't these functions do? Let us learn more here.

Question

If $f (x) = - 1 + | x - 1 |, - 1 \leq x \leq 3; g (x) = 2 - | x + 1 |, - 2 \leq x \leq 2$ , then calculate $(f o g) (x)$ & $(g o f) (x)$ . Draw their graph. Discuss the continuity of $(f o g) (x)$ at $x = - 1$ & the differentiability of $(g o f) (x)$ at $x = 1$ .

Accepted Answer

$f (g (x)) = - 1 + |g (x) - 1|, - 1 \leq g (x) \leq 3, - 2 \leq x \leq 2$

$= - 1 + |2 - |x + 1| - 1|, - 1 \leq 2 - |x + 1| \leq 3, - 2 \leq x \leq 2$

$x = 1$

Now, $- 1 \leq 2 - |x + 1| \leq 3$

$= - 3 \leq - |x + 1| \leq 1$

$= - 1 \leq |x + 1| \leq 3$

$\Rightarrow 0 \leq |x + 1| \leq 3 = - 3 \leq x + 1 \leq 3$

$= - 4 \leq x \leq 2$

Also, $- 2 \leq x \leq 2$

$f (g (x)) = - 1 + |1 - |x + 1||, - 2 \leq x \leq 2$

$g (f (x)) = 2 - |f (x) + 1|, - 2 \leq f (x) \leq 2, - 1 \leq x \leq 3$

$= 2 - |- 1 + |x - 1|| + 1, - 2 \leq - 1 + |x - 1| \leq 2, - 1 \leq x \leq 3$

$= 2 - |x - 1|$

Now, $- 2 \leq - 1 + |x - 1| \leq 2 = - 1 \leq |x - 1| \leq 3$

$\Rightarrow - 3 \leq x - 1 \leq 3$

$\Rightarrow - 2 \leq x \leq 4$

Also, $- 1 \leq x \leq 3$

$g (f (x)) = 2 - |x - 1|, - 1 \leq x \leq 3$

$(f o g) (x) = x + 1$ for $- 2 \leq x \leq - 1, - (x + 1)$ for $- 1 < x \leq 0$ and $x - 1$ for $0 < x \leq 2$ . $(f o g) (x) is continuous at x = - 1,$ $(g o f) (x) = x + 1 for - 1 \leq x \leq 1 and 3 - x for 1 < x \leq 3 . (g o f) (x) is not differentiable at x = 1$

If $f (x) = - 1 + | x - 1 |, - 1 \leq x \leq 3; g (x) = 2 - | x + 1 |, - 2 \leq x \leq 2$ , then calculate $(f o g) (x)$ & $(g o f) (x)$ . Draw their graph. Discuss the continuity of $(f o g) (x)$ at $x = - 1$ & the differentiability of $(g o f) (x)$ at $x = 1$ .

Important Questions on Functions

Learn and Prepare for any exam you want !

If f(x)=-1+|x-1|,-1≤x≤3;g(x)=2-|x+1|,-2≤x≤2, then calculate (fog)x & (gof)(x). Draw their graph. Discuss the continuity of (fog)x at x=-1 & the differentiability of (gof)x at x=1.

Important Questions on Functions

Learn and Prepare for any exam you want !

If $f (x) = - 1 + | x - 1 |, - 1 \leq x \leq 3; g (x) = 2 - | x + 1 |, - 2 \leq x \leq 2$ , then calculate $(f o g) (x)$ & $(g o f) (x)$ . Draw their graph. Discuss the continuity of $(f o g) (x)$ at $x = - 1$ & the differentiability of $(g o f) (x)$ at $x = 1$ .