Mathematics

Problem

Consider the real–valued function

f(x)=\dfrac{1}{x},\;x\neq0

. Answer the following questions in detail. a. (i) State the domain and the range of

f

. (2) b. (ii) Give a clear reason why the graph of

f

can never meet either coordinate axis. (2) c. (iii) Describe the behaviour of

f(x)

x\to0^{+},\;x\to0^{-}

and as

|x|\to\infty

. (3) d. (iv) Into how many separate branches is the graph split? Specify the quadrant(s) in which each branch lies and justify your answer. (3)

Intuitive explanation

The core concept here is understanding how a reciprocal function behaves—how it responds to changes in its input and what that tells us about its properties. The key insight is that $f(x) = \frac{1}{x}$ is fundamentally about inverse relationships: as $x$ grows, $f(x)$ shrinks, and vice versa. This symmetry, combined with the function's discontinuity at $x=0$ , creates two separate branches with distinct behaviors worth exploring carefully.

Intuitive explanation

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