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 $1-3$. $$f(x)=\frac{2}{x}$$

Key Concepts

Function Transformations

This concept refers to the process of modifying a basic function to obtain a new function by applying operations such as stretching, shrinking, reflecting, or translating. In the given problem, multiplying the function y = 1/x by a constant (in this case 2) results in a vertical stretch, meaning every y-value of the original function is doubled, while the overall shape and asymptotes remain the same.

Reciprocal Function Properties

Reciprocal functions, such as y = 1/x, are characterized by their hyperbolic curves, with distinct branches located in opposite quadrants. They typically have a vertical asymptote at x = 0 and a horizontal asymptote at y = 0, features that persist even when the function is transformed by a scalar multiplier, such as in f(x) = 2/x.

Domain and Range

The domain of a function is the set of all x-values for which the function is defined, and the range is the set of all resulting y-values. Understanding these concepts is essential when dealing with functions like f(x) = 2/x, where certain x-values (specifically x = 0) must be excluded to prevent division by zero, and the output values are influenced by the behavior of the reciprocal function.

Monotonicity (Increasing/Decreasing Intervals)

Monotonicity describes intervals where a function is consistently increasing or decreasing. For a function like f(x) = 2/x, analyzing how the function behaves in relation to changes in x (either algebraically or via a derivative) helps identify the intervals over which the function is rising or falling, which is important for understanding its overall behavior.

Graphing Rational Functions

Graphing rational functions involves recognizing key features such as asymptotes, intercepts, and the effects of transformations. For functions derived from basic rational functions like y = 1/x, it is important to note how modifications (for example, vertical stretching by a constant multiplier) influence the graph, enabling an accurate sketch of the function's behavior across its domain.

Transcript

Please give Ace some feedback