Discuss the continuity of the function f given by f(x)=x3+x2−1.
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Step-by-Step Solution
Step 1: Identify the function type
The given function is f(x)=x3+x2−1. This is a polynomial function, as it consists of terms involving non-negative integer powers of x combined with addition and subtraction.
Step 2: Recall continuity property of polynomials
A fundamental property of polynomial functions is that they are continuous for all real numbers. This means that for any real number c, the limit of the function as x approaches c exists and is equal to the function's value at c, i.e., limx→cf(x)=f(c).
Step 3: Conclusion on continuity
Since f(x)=x3+x2−1 is a polynomial function, it is continuous for all real numbers. There are no points where the function is undefined or where its graph would have a break.