The function f(x)=√(1-√(1-x^2)) (a) has its domain -1 ≤x ≤1 (b) has finite one sided derivates a

step 1 So in this question we have since, since G of X is E1, E1 and and differentiable at X is equal t

The function f(x)=√(1-√(1-x^2)) (a) has its domain -1 ≤x ≤1...

The function $f(x)=\sqrt{1-\sqrt{1-x^{2}}}$
(a) has its domain $-1 \leq x \leq 1$
(b) has finite one sided derivates at the point $x=0$
(c) is continuous and differentiable at $x=0$
(d) is continuous but not differentiable at $x=0$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Differentiability and One-Sided Derivatives

A function is differentiable at a point if its derivative exists and is the same when approached from both the left and the right. The existence of finite one-sided derivatives indicates that the slopes from each direction are well-defined; however, if these one-sided derivatives are not equal, the function exhibits a corner or cusp at that point, meaning it is not fully differentiable there.

Domain of Square Root Functions

When dealing with functions that involve square roots, the radicand (the expression inside the square root) must be nonnegative to ensure that the function is defined over the real numbers. This principle extends to nested square roots, where each inner expression must independently satisfy the nonnegativity requirement, thereby influencing the overall domain of the function.

Continuity of Functions

Continuity at a point means that the limit of the function as the variable approaches that point equals the function’s value there. This concept is critical for ensuring that the function has no jumps or breaks, thereby exhibiting a smooth transition in its graph.

Transcript

Please give Ace some feedback