Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 6: Problem 45
Use transformations to graph each function. Determine the domain, range, horizontal asymptote, and y-intercept of each function. $$ f(x)=2^{x}+1 $$
Short Answer
Expert verified
Domain: \( (-fty, fty) \). Range: \( (1, fty) \). Horizontal Asymptote: y = 1. Y-intercept: (0, 2).
Step by step solution
01
Understand the parent function
The parent function for this exercise is the exponential function, which is generally written as \[ f(x) = 2^x \]. In this form, the graph of this function passes through the point (0,1), increases rapidly, and has a horizontal asymptote at y=0.
02
Apply the vertical translation
The function given is \[ f(x) = 2^x + 1 \]. This indicates that the graph of the parent function \[ 2^x \] is shifted vertically upwards by 1 unit. This means every point on the graph of \[ 2^x \] will move one unit up.
03
Graph the function
To graph the function \[ f(x) = 2^x + 1 \], start by plotting the key points of the parent function \[ 2^x \], and then shift each point one unit upwards. The new points will include (0, 2), (1, 3), and (-1, 1.5), among others.
04
Determine the domain
The domain of the function \[ f(x) = 2^x + 1 \] is all real numbers because an exponential function is defined for all x. Thus, the domain is \( (-fty, fty) \).
05
Determine the range
Since \[ 2^x \] is always positive and shifts up by 1 unit, the output (range) of \[ 2^x + 1 \] will always be greater than 1. Therefore, the range of the function is \( (1, fty) \).
06
Identify the horizontal asymptote
The horizontal asymptote of the parent function \[ 2^x \] is y = 0. However, shifting this function up by 1 unit changes the horizontal asymptote to y = 1.
07
Find the y-intercept
To find the y-intercept, set \( x = 0 \). \[ f(0) = 2^0 + 1 = 1 + 1 = 2 \]. Therefore, the y-intercept is at the point (0, 2).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
transformations
Transformations help us modify the parent function into a new function by shifting, stretching, or compressing its graph. Here, we focus on vertical translations. The parent function given is an exponential function: \[ f(x) = 2^x \]. This forms the base of our transformations.
When we look at \[ f(x) = 2^x + 1 \], we see that the '+1' term translates the entire graph of \[ 2^x \] one unit upwards.
This is because adding a constant to a function results in shifting the graph vertically. Therefore, every point on \[ 2^x \] moves one unit higher. Understanding transformations like this helps in sketching graphs and predicting their changes accurately.
When we look at \[ f(x) = 2^x + 1 \], we see that the '+1' term translates the entire graph of \[ 2^x \] one unit upwards.
This is because adding a constant to a function results in shifting the graph vertically. Therefore, every point on \[ 2^x \] moves one unit higher. Understanding transformations like this helps in sketching graphs and predicting their changes accurately.
domain
The domain of a function consists of all the input values (x-values) for which the function is defined. In the case of exponential functions like \[ f(x) = 2^x + 1 \], there's no restriction on x.
This means you can plug any real number into the function and get a valid output.
Therefore, the domain remains all real numbers, commonly expressed as \[ (-\fty, \fty) \]. This indicates that the function can take any value of x, from minus infinity to positive infinity.
This means you can plug any real number into the function and get a valid output.
- The exponential function \[ 2^x \] itself is defined for all real numbers.
- The additional '+1' just shifts the output vertically but does not affect the domain.
Therefore, the domain remains all real numbers, commonly expressed as \[ (-\fty, \fty) \]. This indicates that the function can take any value of x, from minus infinity to positive infinity.
range
The range of a function consists of all possible output values (y-values). For \[ f(x) = 2^x + 1 \], the exponential part, \[ 2^x \], always gives positive values. Since we are adding 1 to these values, the output will always be greater than 1.
Therefore, the range excludes 1 but includes every y-value greater than 1.
So, the range of \[ f(x) = 2^x + 1 \] is \[ (1, \fty) \]. This specifies that the function outputs values starting just above 1 and continuing towards infinity.
Therefore, the range excludes 1 but includes every y-value greater than 1.
- Initially, for \[ 2^x \], the range is (0, \[ \fty \]).
- Adding 1 shifts every output up by 1 unit, making the range (1, \[ \fty \]).
So, the range of \[ f(x) = 2^x + 1 \] is \[ (1, \fty) \]. This specifies that the function outputs values starting just above 1 and continuing towards infinity.
horizontal asymptote
To find the horizontal asymptote of an exponential function like \[ f(x) = 2^x + 1 \], observe where the function stabilizes as x approaches infinity or negative infinity.
For the parent function \[ 2^x \], the horizontal asymptote is y = 0 because no matter how large or small x gets, the function never touches y = 0. When we add 1 to this, we shift the entire graph up by 1 unit.
This effectively moves the horizontal asymptote from y = 0 to y = 1.
The horizontal asymptote of \[ f(x) = 2^x + 1 \] is therefore y = 1. This is where the function levels off but never actually reaches.
For the parent function \[ 2^x \], the horizontal asymptote is y = 0 because no matter how large or small x gets, the function never touches y = 0. When we add 1 to this, we shift the entire graph up by 1 unit.
This effectively moves the horizontal asymptote from y = 0 to y = 1.
- For \[ 2^x \], asymptote y = 0.
- After the shift (adding +1), asymptote y = 1.
The horizontal asymptote of \[ f(x) = 2^x + 1 \] is therefore y = 1. This is where the function levels off but never actually reaches.
y-intercept
The y-intercept is the point where the graph crosses the y-axis. To find this, set x to 0 in the function \[ f(x) = 2^x + 1 \].
For \[ x = 0 \], the computation is simple:
\[ f(0) = 2^0 + 1 = 1 + 1 = 2 \].
So, the y-intercept is the point where y equals 2 when x is 0. You can denote this as the coordinate pair (0, 2).
In summary, the y-intercept for \[ f(x) = 2^x + 1 \] is at (0, 2).
For \[ x = 0 \], the computation is simple:
\[ f(0) = 2^0 + 1 = 1 + 1 = 2 \].
So, the y-intercept is the point where y equals 2 when x is 0. You can denote this as the coordinate pair (0, 2).
- The y-intercept shows where the graph begins on the y-axis.
In summary, the y-intercept for \[ f(x) = 2^x + 1 \] is at (0, 2).
