Graph x^5

Problem

ƒ(x)=x5

Solution

1. Identify the function as an odd power function of the form ƒ(x)=xn where n=5

2. Determine symmetry by checking ƒ*(−x)=(−x)5=−x5 which confirms the graph has origin symmetry (it is an odd function).

3. Find the intercepts by evaluating ƒ(0)=0=0 showing the graph passes through the origin (0,0)

4. Analyze end behavior as x→∞ ƒ(x)→∞ and as x→−∞ ƒ(x)→−∞

5. Plot key points such as (−1,−1) (1,1) (−2,−32) and (2,32) to determine the steepness of the curve.

6. Observe the shape near the origin, where the graph is flatter than y=x3 because |x|5<|*x|3 for 0<|x|<1

Final Answer

ƒ(x)=x5

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