Analyze the function f(x) = 1/x: Domain, Range, Intercepts, Asymptotes, and Graph

f(x)=(1/x)

To analyze the function f(x) = 1/x, there are a few important components to consider: the domain, range, intercepts, asymptotes, and a graph

To analyze the function f(x) = 1/x, there are a few important components to consider: the domain, range, intercepts, asymptotes, and a graph.

1. Domain: The domain refers to the set of all possible input values (x-values) for a function. For this function, we need to exclude any values of x that would result in division by zero, since division by zero is undefined. Therefore, the domain of f(x) = 1/x is all real numbers except for x = 0. In interval notation, the domain is (-∞, 0) U (0, ∞).

2. Range: The range represents the set of all possible output values (or y-values) for a function. Since the function f(x)=1/x can take any positive or negative value (excluding zero), the range is (-∞, 0) U (0, ∞).

3. Intercepts: To find the x-intercept, set f(x) equal to zero and solve for x:
0 = 1/x
This equation has no solution since there is no value of x that makes the division by zero equal to zero. Therefore, this function has no x-intercept.

To find the y-intercept, set x equal to zero:
f(0) = 1/0 (undefined)
Since the value of f(0) is undefined, this function has no y-intercept.

4. Asymptotes: Asymptotes refer to the lines that the graph of the function approaches but does not cross. There are two types of asymptotes to consider for this function: vertical asymptotes and horizontal asymptotes.

Vertical asymptote: A vertical asymptote occurs when the function approaches positive or negative infinity as x approaches a particular value. For f(x) = 1/x, there is a vertical asymptote at x = 0, since as x approaches 0 from either side, the function approaches positive or negative infinity.

Horizontal asymptote: A horizontal asymptote occurs when the function approaches a particular value as x approaches positive or negative infinity. For f(x) = 1/x, there is a horizontal asymptote at y = 0, since as x approaches infinity or negative infinity, the function approaches 0.

5. Graph: The graph of f(x) = 1/x is a curve that starts at positive infinity as x approaches negative infinity, passes through (1,1), and continues towards zero as x approaches positive infinity. The graph approaches the vertical asymptote at x = 0 and the horizontal asymptote at y = 0.

Here is a graphical representation of f(x) = 1/x:

/
/
/|
|\
| \
| \
——

Note that the graph will never actually touch the asymptotes, but it will approach them as x gets closer to zero.

More Answers: