Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Get expert answers to your questions quickly and accurately from our dedicated community of professionals. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To determine which function has a horizontal asymptote of [tex]\( y = 3 \)[/tex], we need to carefully analyze each function and how they behave as [tex]\( x \)[/tex] approaches infinity ([tex]\( +\infty \)[/tex]) or negative infinity ([tex]\( -\infty \)[/tex]).
1. Function: [tex]\( f(x) = 3 \cdot 2^x \)[/tex]
As [tex]\( x \)[/tex] approaches infinity:
[tex]\[ 2^x \][/tex] grows exponentially to infinity, and so does [tex]\( 3 \cdot 2^x \)[/tex].
As [tex]\( x \)[/tex] approaches negative infinity:
[tex]\[ 2^x \][/tex] approaches 0, making [tex]\( 3 \cdot 2^x \)[/tex] also approach 0.
So, the horizontal asymptote is [tex]\( y = 0 \)[/tex].
2. Function: [tex]\( f(x) = 2 \cdot 4^{x-3} \)[/tex]
As [tex]\( x \)[/tex] approaches infinity:
Since [tex]\( 4^{x-3} \)[/tex] is an exponential function, it will grow to infinity, and so will [tex]\( 2 \cdot 4^{x-3} \)[/tex].
As [tex]\( x \)[/tex] approaches negative infinity:
[tex]\[ 4^{x-3} \][/tex] approaches 0, making [tex]\( 2 \cdot 4^{x-3} \)[/tex] also approach 0.
So, the horizontal asymptote is [tex]\( y = 0 \)[/tex].
3. Function: [tex]\( f(x) = 2 \cdot 3^x \)[/tex]
As [tex]\( x \)[/tex] approaches infinity:
[tex]\[ 3^x \][/tex] grows exponentially to infinity, and so does [tex]\( 2 \cdot 3^x \)[/tex].
As [tex]\( x \)[/tex] approaches negative infinity:
[tex]\[ 3^x \][/tex] approaches 0, making [tex]\( 2 \cdot 3^x \)[/tex] also approach 0.
So, the horizontal asymptote is [tex]\( y = 0 \)[/tex].
4. Function: [tex]\( f(x) = 2 \cdot 4^x + 3 \)[/tex]
As [tex]\( x \)[/tex] approaches infinity:
[tex]\[ 4^x \][/tex] grows exponentially to infinity, and so does [tex]\( 2 \cdot 4^x \)[/tex]. Hence, [tex]\( f(x) \)[/tex] also grows to infinity.
As [tex]\( x \)[/tex] approaches negative infinity:
[tex]\[ 4^x \][/tex] approaches 0. Thus, [tex]\( 2 \cdot 4^x \)[/tex] also approaches 0 making the function [tex]\( f(x) = 0 + 3 = 3 \)[/tex].
So, the horizontal asymptote is [tex]\( y = 3 \)[/tex].
From this analysis, we see that the function [tex]\( f(x) = 2(4^x) + 3 \)[/tex] has a horizontal asymptote of [tex]\( y = 3 \)[/tex].
Therefore, the function with a horizontal asymptote of [tex]\( y = 3 \)[/tex] is:
[tex]\[ f(x) = 2 \left(4^x\right) + 3 \][/tex]
1. Function: [tex]\( f(x) = 3 \cdot 2^x \)[/tex]
As [tex]\( x \)[/tex] approaches infinity:
[tex]\[ 2^x \][/tex] grows exponentially to infinity, and so does [tex]\( 3 \cdot 2^x \)[/tex].
As [tex]\( x \)[/tex] approaches negative infinity:
[tex]\[ 2^x \][/tex] approaches 0, making [tex]\( 3 \cdot 2^x \)[/tex] also approach 0.
So, the horizontal asymptote is [tex]\( y = 0 \)[/tex].
2. Function: [tex]\( f(x) = 2 \cdot 4^{x-3} \)[/tex]
As [tex]\( x \)[/tex] approaches infinity:
Since [tex]\( 4^{x-3} \)[/tex] is an exponential function, it will grow to infinity, and so will [tex]\( 2 \cdot 4^{x-3} \)[/tex].
As [tex]\( x \)[/tex] approaches negative infinity:
[tex]\[ 4^{x-3} \][/tex] approaches 0, making [tex]\( 2 \cdot 4^{x-3} \)[/tex] also approach 0.
So, the horizontal asymptote is [tex]\( y = 0 \)[/tex].
3. Function: [tex]\( f(x) = 2 \cdot 3^x \)[/tex]
As [tex]\( x \)[/tex] approaches infinity:
[tex]\[ 3^x \][/tex] grows exponentially to infinity, and so does [tex]\( 2 \cdot 3^x \)[/tex].
As [tex]\( x \)[/tex] approaches negative infinity:
[tex]\[ 3^x \][/tex] approaches 0, making [tex]\( 2 \cdot 3^x \)[/tex] also approach 0.
So, the horizontal asymptote is [tex]\( y = 0 \)[/tex].
4. Function: [tex]\( f(x) = 2 \cdot 4^x + 3 \)[/tex]
As [tex]\( x \)[/tex] approaches infinity:
[tex]\[ 4^x \][/tex] grows exponentially to infinity, and so does [tex]\( 2 \cdot 4^x \)[/tex]. Hence, [tex]\( f(x) \)[/tex] also grows to infinity.
As [tex]\( x \)[/tex] approaches negative infinity:
[tex]\[ 4^x \][/tex] approaches 0. Thus, [tex]\( 2 \cdot 4^x \)[/tex] also approaches 0 making the function [tex]\( f(x) = 0 + 3 = 3 \)[/tex].
So, the horizontal asymptote is [tex]\( y = 3 \)[/tex].
From this analysis, we see that the function [tex]\( f(x) = 2(4^x) + 3 \)[/tex] has a horizontal asymptote of [tex]\( y = 3 \)[/tex].
Therefore, the function with a horizontal asymptote of [tex]\( y = 3 \)[/tex] is:
[tex]\[ f(x) = 2 \left(4^x\right) + 3 \][/tex]
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.