Graph x^5

Problem

ƒ⁡(x)=x5 f with hook of x equals x to the fifth power

Solution

  1. Identify the function as an odd power function of the form ƒ⁡(x)=xn f with hook of x equals x to the n-th power where n=5 n equals 5

  2. Determine symmetry by checking ƒ⁢(−x)=(−x)5=−x5 f with hook times negative x equals open paren negative x close paren to the fifth power equals negative x to the fifth power which confirms the graph has origin symmetry (it is an odd function).

  3. Find the intercepts by evaluating ƒ⁡(0)=05=0 f with hook of 0 equals 0 to the fifth power equals 0 showing the graph passes through the origin (0,0) open paren 0 comma 0 close paren

  4. Analyze end behavior as x→∞ x right arrow infinity ƒ⁡(x)→∞ f with hook of x right arrow infinity and as x→−∞ x right arrow negative infinity ƒ⁡(x)→−∞ f with hook of x right arrow negative infinity

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

  6. Observe the shape near the origin, where the graph is flatter than y=x3 y equals x cubed because |x|5<|⁢x|3 the absolute value of x to the fifth power is less than vertical bar x vertical bar cubed for 0<|x|<1 0 is less than the absolute value of x is less than 1

Final Answer

ƒ⁡(x)=x5 f with hook of x equals x to the fifth power

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