Explain how the graph of each function can be obtained from...

Explain how the graph of each function can be obtained from the graph of $y=\frac{1}{x}$ or $y=\frac{1}{x^{2}} .$ Then graph $f$ and give the (a) domain and (b) range. Determine the intervals of the domain for which the function is ( $c$ ) increasing or (d) decreasing. See Examples $1-3$. $$f(x)=-\frac{2}{x^{2}}$$

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Key Concepts

Graph Transformations

This concept involves obtaining the graph of a function by applying modifications such as scaling, reflecting, and translating, to a basic or parent function. By understanding how a function like y = 1/x² behaves, one can apply transformations (in this case, a vertical stretch by a factor of 2 and a reflection across the x-axis) to derive the graph of a function like -2/x².

Domain and Range

The domain of a function is the set of all allowable inputs, while the range is the set of all possible outputs. Determining these involves understanding the restrictions imposed by the function’s algebraic expression (for instance, the denominator in a rational function cannot be zero) and the effect of any transformations, such as reflections or vertical stretches, on the output values.

Monotonicity (Increasing/Decreasing Intervals)

Monotonicity refers to the behavior of a function in terms of whether it is increasing or decreasing over certain intervals. This can be examined by analyzing the behavior of the function or its derivative. If a function’s derivative is positive over an interval, the function is increasing there; if negative, it is decreasing. This analysis helps in identifying the intervals of the domain for which the function’s output values consistently rise or fall.

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