f(x):

𝑓
(
𝑥
)
=
{
2
−
∣
𝑥
+
1
∣
,
−
4
≤
𝑥
<
0
𝑥
,
0
≤
𝑥
≤
4
f(x)={ 
2−∣x+1∣,
sqrt x
​
 ,
​
  
−4≤x<0
0≤x≤4
​
 
𝑔
(
𝑥
)
g(x):

𝑔
(
𝑥
)
=
{
2
∣
𝑥
∣
,
−
1
≤
𝑥
<
1
3
−
𝑥
,
1
≤
𝑥
<
3
(
𝑥
−
3
)
2
,
3
≤
𝑥
≤
4
g(x)= 
⎩
⎨
⎧
​
  
2∣x∣,
3−x,
(x−3) 
2
 ,
​
  
−1≤x<1
1≤x<3
3≤x≤4    give graph of fof and gog 
​
 

Here are the graphs of the compositions \( f(f(x)) \) and \( g(g(x)) \). The first graph shows \( f(f(x)) \), and the second shows \( g(g(x)) \), each plotted according to the domains defined in the piecewise functions.

Would you like a detailed explanation of the steps involved, or do you have any questions?

Here are 5 related questions to consider:
1. How do we interpret piecewise functions in graphing?
2. What is the role of absolute values in defining \( f(x) \) and \( g(x) \)?
3. How would we handle discontinuities in these piecewise functions?
4. What happens to the composition \( f(f(x)) \) or \( g(g(x)) \) outside their respective domains?
5. How could you modify these functions to shift the graphs horizontally or vertically?

**Tip**: Always check the domain of piecewise functions when composing them to avoid errors in undefined regions.

The query submitted by the user: f(x) = { 2 − |x+1|, −4 ≤ x < 0 x, 0 ≤ x ≤ 4 } and g(x) = { 2|x|, −1 ≤ x < 1 3 − x, 1 ≤ x < 3 (x − 3)^2, 3 ≤ x ≤ 4 } Give the graph of f(f(x)) and g(g(x)).

Explore how to graph the compositions f(f(x)) and g(g(x)) for the given piecewise functions. Learn about the key concepts like absolute value, function composition, and the domain of functions, with step-by-step visualizations and solutions. Ideal for high school students in Grades 10-12.

Math Problem Statement

𝑓 ( 𝑥 )

𝑔 ( 𝑥 )

Solution

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