If f(x)={ -1: x 0 . then

step 1 Hello everyone we are going to solve a question in this question we are given two functions f x

If f(x)={ -1: x<0 0 : x=0 and ...

If $f(x)=\left\{\begin{array}{lll}-1: & x<0 \\ 0 & : & x=0 \text { and } g(x)=x\left(1-x^{2}\right) \\ 1 & : & x>0\end{array}\right.$
then discuss the continuity of the function $h(x)$, where $h(x)=f(g(x))$.

Key Concepts

Zero Analysis in Functional Composition

Zero analysis involves finding the zeros of a function, which are the inputs where the function's output is zero. In the context of composite functions where the outer function depends on the sign of its input (e.g., being defined differently for negative, zero, or positive values), locating the zeros of the inner function is essential. These zeros can indicate where the composite function might change its behavior, possibly leading to discontinuities if the function's value does not transition smoothly.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain. These functions often have potential discontinuities at the boundaries between pieces. When a piecewise function is used as part of a composite function, the transition points where the definition changes are key areas to analyze for possible jump discontinuities or other forms of discontinuity.

Composite Functions

Composite functions are formed when one function is applied to the result of another function. Understanding composite functions involves analyzing how the properties, such as continuity, of the inner function affect the overall behavior when passed through the outer function. In this context, the continuity of the composed function is examined by determining if the inner function's outputs are mapped seamlessly by the outer function without jumps or gaps.

Continuity of Functions

Continuity of a function at a point means that the function's value at that point matches the limit of the function as the variable approaches that point. When assessing the continuity of a composite function, it is critical to examine the continuity of both the inner and outer functions and to pay special attention to points where the inner function's output causes a transition in the behavior of the outer function.

Transcript

Please give Ace some feedback