Graph each function. See Examples 1 and 2 . f(x)=((1)/(5))^x...

Name: Problem 24, Chapter 11: Elementary and Intermediate Algebra
Uploaded: 2020-06-25T18:27:14-08:00
Duration: 4 min 54 s
Channel: Nick Johnson
Description: Graph each function. See Examples 1 and 2 . f(x)=((1)/(5))^x

Key Concepts

Exponential Functions

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent, typically expressed in the form f(x) = a^x. This type of function is fundamental in describing growth or decay processes in various real-world situations. The behavior of the function largely depends on the base; if the base is greater than 1, the function exhibits exponential growth, while if it is between 0 and 1, the function shows exponential decay.

Graphing Exponential Functions

Graphing exponential functions involves plotting the function by calculating values for various inputs and observing the overall trend of the curve. These graphs are characterized by a rapid increase or decrease depending on the base of the exponent. Key features such as the y-intercept and the horizontal asymptote are identified to help understand the behavior of the function across the domain.

Exponential Decay

Exponential decay occurs in exponential functions where the base is a fraction between 0 and 1, leading to a decrease in function values as the independent variable increases. This behavior is common in natural and physical phenomena where a quantity diminishes over time, such as radioactive decay or cooling processes.

Asymptotic Behavior

A key characteristic of exponential functions is their asymptotic nature. For functions of the form f(x) = a^x, the graph will approach but never actually cross the horizontal line known as the asymptote. In the case of exponential decay, the horizontal asymptote is typically the x-axis (y = 0), indicating that the function values get arbitrarily close to zero as x increases.

Domain and Range

Exponential functions have a domain of all real numbers and a range of positive real numbers. This means that while the function can accept any real number as an input, the output or the function value will always be positive, never reaching zero or becoming negative. This property is essential when evaluating and graphing these functions.

Transcript

00:01 So here we are given the function f of x is equal to one -fifth to the x.

00:14 And we want to graph this on our x, y, axes here.

00:18 So if we were given some points on our function, we could then just connect those points, and we have our graph.

00:27 So a way to do that is just to substitute in some values into our function.

00:33 For x and then the output the f of x will be the y point so if we give let's say we start at um x is zero when x is zero what is f of x when x is zero we have one -fifth to the zero what is that well anything right anything other than maybe zero anything to the zero power is equal to one so when x is zero f of x is one and f of x is one and f of x is just another name for y so we have um when x is zero y is one all right how about when x is one or when x is one we have one fifth to the one one fifth to the one is just one fifth so the point one one fifth is on our graph when x is one y or f of x is equal to one fifth um if we did x is two well we'd have one -fifth to the two, or one -fifth squared.

01:38 So we square the numerator and square the denominator, but the numerator is one.

01:43 So one to any power is just one, and five squared is 25.

01:47 So one -fifth squared is one -twenty -fifth.

01:51 So when x is two, y is one -twentifth.

01:57 How about some negative values? well, let's say when x is negative 1, what is f of x? well, then we'd have 1 fifth to the negative 1.

02:08 Remember what a negative exponent means.

02:10 That means 1 over 1 5th to the 1.

02:14 Well, 1 over 1 5th just takes 1 5th and basically flips it upside down or takes the reciprocal.

02:21 So 1 5 to the negative 1 is just 5 over 1, which is just 5.

02:27 So when x is negative 1, f of x is a positive 5.

02:33 And when x, let's say, is negative 2, well, then we would have 1 5th to the negative 2.

02:41 So that's 1 over 1 5th to the 2, or 1 over, or 1 over, well, 1 over 1 5th, again, that takes 1 5th and flips upside down.

02:52 So 1 over 1 5 squared is the same thing as 5 squared.

02:58 Because 1 over 1 5th is 5 over 1...

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