[tex]f(x)=\frac{x^2-4}{x+2}[/tex]
Answer (2)
Factor the numerator: x 2 − 4 = ( x − 2 ) ( x + 2 ) .
Rewrite the function: f ( x ) = x + 2 ( x − 2 ) ( x + 2 ) .
Simplify the function, noting the restriction x = − 2 : f ( x ) = x − 2 for x = − 2 .
Identify the hole in the graph at x = − 2 , which is the point ( − 2 , − 4 ) .
Explanation
Understanding the Function We are given the function f ( x ) = x + 2 x 2 − 4 and we want to analyze it. This involves simplifying the function and identifying any points where the function is not defined.
Factoring the Numerator First, we factor the numerator of the function. The numerator is a difference of squares, so we can factor it as follows: x 2 − 4 = ( x − 2 ) ( x + 2 ) Thus, we can rewrite the function as: f ( x ) = x + 2 ( x − 2 ) ( x + 2 )
Simplifying the Function Next, we simplify the function by canceling the common factor ( x + 2 ) in the numerator and the denominator. However, we must note that this cancellation is only valid when x + 2 = 0 , which means x = − 2 . So, for x = − 2 , we have: f ( x ) = x − 2
Identifying the Hole The simplified function is f ( x ) = x − 2 , but we must remember the restriction that x = − 2 . This means that there is a hole in the graph of the function at x = − 2 . To find the y-coordinate of the hole, we substitute x = − 2 into the simplified function: y = ( − 2 ) − 2 = − 4 So, the hole is at the point ( − 2 , − 4 ) .
Final Answer In conclusion, the function f ( x ) = x + 2 x 2 − 4 is equivalent to the line y = x − 2 with a hole at the point ( − 2 , − 4 ) .
Examples
Understanding functions with holes is crucial in various fields. For instance, in physics, when modeling the trajectory of a projectile, certain conditions might lead to undefined points in the equation, representing physical limitations or singularities. Similarly, in economics, when analyzing cost functions, a hole might indicate a point where the model breaks down due to unforeseen circumstances, such as a sudden market crash or a supply chain disruption. Recognizing and interpreting these 'holes' allows for more accurate and robust modeling in real-world scenarios.
The function f ( x ) = x + 2 x 2 − 4 simplifies to f ( x ) = x − 2 for x = − 2 , creating a hole at the point ( − 2 , − 4 ) . To find this hole, factor the numerator and cancel out the common term in the denominator. The hole represents a point where the function is not defined even though the simplified version implies a linear relationship.
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