Show that f(x)={ x^2+1, x ≤ 1 2 x, x>1 . is continuous and differentiable at x

step 1 Alright, so question 43 gives us a compound function, x squared plus 1, where x is less than equ

Show that f(x)={ x^2+1, x ≤ 1 2 x, x>1 ....

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Key Concepts

Piecewise-defined Function

A piecewise-defined function is one that is described by different expressions over different intervals of its domain. This requires special attention at the boundary points where the definition changes, ensuring proper evaluation for continuity and differentiability.

Continuity at a Point

Continuity at a specific point means that the function does not have any breaks or jumps at that point. Specifically, the limit of the function as it approaches the point from the left must equal the limit from the right, and both must equal the function's value at that point.

Differentiability at a Point

Differentiability at a point means that the function has a well-defined tangent at that point, i.e., the derivative exists there. For piecewise functions, this requires that the derivative calculated from each side of the point is the same, ensuring a smooth transition.

Graphing of Functions

Graphing a function provides a visual representation of its behavior. With piecewise-defined functions, it is important to correctly plot each segment of the function and indicate whether endpoints are included or excluded, especially at the transition points, to accurately reflect the function's continuity and differentiability.

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