Let f: R →R be any function. Define g: R →R by g(x) =|f(x)| for all x. Then g is (A) onto if f i

step 1 In this problem, let us assume that we have an HX which is equals to mod X for all X belonging t

Let f: R →R be any function. Define g: R →R by g(x) =|f(x)|...

Let $f: R \rightarrow R$ be any function. Define $g: R \rightarrow R$ by $g(x)$ $=|f(x)|$ for all $x$. Then $g$ is
(A) onto if $f$ is onto
(B) one-one if $f$ is one-one
(C) continuous if $f$ is continuous
(D) differentiable if $f$ is differentiable

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Key Concepts

Surjectivity

Surjectivity, or onto property, of a function, means that every element in the codomain is the image of at least one element from the domain. In the context of the problem, the operation of taking the absolute value restricts the output range to nonnegative values, which may affect surjectivity even if the original function was surjective.

Injectivity

Injectivity, or one-to-one property, means that different elements in the domain map to different elements in the codomain. When applying the absolute value, the fact that f(x) and -f(x) yield the same value for g(x) can cause the function to lose injectivity, unless additional constraints are imposed.

Continuity

Continuity of a function implies that small changes in the input lead to small changes in the output, without abrupt jumps or breaks. When a continuous function is composed with the absolute value function, the resulting function remains continuous, because the absolute value function itself is continuous.

Differentiability

Differentiability means that a function has a derivative at every point in its domain. Even if a function is differentiable, applying the absolute value can cause nondifferentiable points (for instance, at points where the original function crosses zero) because the absolute value function has a 'corner' at zero, which can disrupt smoothness.

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