Q. 9 - Precalculus Enhanced with Graphing Utilities

Question

Graph $f\left(x\right)=3^{x-2}+1$ using transformations. What is the domain, range, and horizontal asymptote of $f$?

Step-by-Step Solution

Answer

The graph of the function $f\left(x\right)=3^x$ is,

The graph of the function $f\left(x\right)=3^{x-2}$ is, 

The graph of the function $f\left(x\right)=3^{x-2}+1$ is,

The domain of the function $f\left(x\right)=3^{x-2}+1$ is, the set of all the real numbers, the range of the given function is,$\left\{y|y>1\right\}$ and the horizontal asymptote is $y=1$.

Step 1 Given function is,

$f\left(x\right)=3^{x-2}+1$.

First draw a table for $f\left(x\right)=3^x$ and sketch a graph for it.

Step 2 Now plot a graph for it.

Step 3 Replace x with x - 2 on the right side of src="data:image/svg+xml;base64,<svg xmlns="http://www.w3.org/2000/svg" xmlns:wrs="http://www.wiris.com/xml/mathml-extension" height="23" width="62" wrs:baseline="18"><!--MathML: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mn>3</mn><mi>x</mi></msup></math>--><defs><style type="text/css">@font-face{font-family:'aec8956637a99787bd197eacd77acce';src:url(data:font/truetype;charset=utf-8;base64,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)format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'math17f39f8317fbdb1988ef4c628eb';src:url(data:font/truetype;charset=utf-8;base64,AAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAADRjdnQgDVUNBwAAAVAAAAA6Z2x5ZoPi2VsAAAGMAAAAsmhlYWQQC2qxAAACQAAAADZoaGVhCGsXSAAAAngAAAAkaG10eE2rRkcAAAKcAAAACGxvY2EAHTwYAAACpAAAAAxtYXhwBT0FPgAAArAAAAAgbmFtZaBxlY4AAALQAAABn3Bvc3QB9wD6AAAEcAAAACBwcmVwa1uragAABJAAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACAAAAAEAAQAAQAAAD3//wAAAD3////EAAEAAAAAAAABVAMsAIABAABWACoCWAIeAQ4BLAIsAFoBgAKAAKAA1ACAAAAAAAAAACsAVQCAAKsA1QEAASsABwAAAAIAVQAAAwADqwADAAcAADMRIRElIREhVQKr/asCAP4AA6v8VVUDAAACAIAA6wLVAhUAAwAHAGUYAbAIELAG1LAGELAF1LAIELAB1LABELAA1LAGELAHPLAFELAEPLABELACPLAAELADPACwCBCwBtSwBhCwB9SwBxCwAdSwARCwAtSwBhCwBTywBxCwBDywARCwADywAhCwAzwxMBMhNSEdASE1gAJV/asCVQHAVdVVVQAAAAEAAAABAADVeM5BXw889QADBAD/////1joTc//////WOhNzAAD/IASAA6sAAAAKAAIAAQAAAAAAAQAAA+j/agAAF3AAAP+2BIAAAQAAAAAAAAAAAAAAAAAAAAIDUgBVA1YAgAAAAAAAAAAoAAAAsgABAAAAAgBeAAUAAAAAAAIAgAQAAAAAAAQAAN4AAAAAAAAAFQECAAAAAAAAAAEAEgAAAAAAAAAAAAIADgASAAAAAAAAAAMAMAAgAAAAAAAAAAQAEgBQAAAAAAAAAAUAFgBiAAAAAAAAAAYACQB4AAAAAAAAAAgAHACBAAEAAAAAAAEAEgAAAAEAAAAAAAIADgASAAEAAAAAAAMAMAAgAAEAAAAAAAQAEgBQAAEAAAAAAAUAFgBiAAEAAAAAAAYACQB4AAEAAAAAAAgAHACBAAMAAQQJAAEAEgAAAAMAAQQJAAIADgASAAMAAQQJAAMAMAAgAAMAAQQJAAQAEgBQAAMAAQQJAAUAFgBiAAMAAQQJAAYACQB4AAMAAQQJAAgAHACBAE0AYQB0AGgAIABGAG8AbgB0AFIAZQBnAHUAbABhAHIATQBhAHQAaABzACAARgBvAHIAIABNAG8AcgBlACAATQBhAHQAaAAgAEYAbwBuAHQATQBhAHQAaAAgAEYAbwBuAHQAVgBlAHIAcwBpAG8AbgAgADEALgAwTWF0aF9Gb250AE0AYQB0AGgAcwAgAEYAbwByACAATQBvAHIAZQAAAwAAAAAAAAH0APoAAAAAAAAAAAAAAAAAAAAAAAAAALkHEQAAjYUYALIAAAAVFBOxAAE/)format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'round_brackets18549f92a457f2409';src:url(data:font/truetype;charset=utf-8;base64,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)format('truetype');font-weight:normal;font-style:normal;}</style></defs><text font-family="aec8956637a99787bd197eacd77acce" font-size="16" font-style="italic" text-anchor="middle" x="3.5" y="18">f</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="12.5" y="18">(</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="25.5" y="18">)</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="18.5" y="18">x</text><text font-family="math17f39f8317fbdb1988ef4c628eb" font-size="16" text-anchor="middle" x="36.5" y="18">=</text><text font-family="Arial" font-size="16" text-anchor="middle" x="49.5" y="18">3</text><text font-family="Arial" font-size="12" font-style="italic" text-anchor="middle" x="57.5" y="11">x</text></svg>" role="math" localid="1647080090238" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/9d265a9c-82c1-47cc-ab09-8507fbbff075.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220312%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220312T104303Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=5925940b576ed0728bc788ba8c20ff6d905a97c70722bfe91f784ce04ba9edf2" f x = 3 x .

So, the obtained function is, $f\left(x\right)=3^{x-2}$.

The graph of $f\left(x\right)=3^{x-2}$ is obtained by horizontally shifting the graph of $f\left(x\right)=3^x$to the right by $2$ units.

Now sketch a graph for the function $f\left(x\right)=3^{x-2}$.

Step 4 Add 1 to the right of src="data:image/svg+xml;base64,<svg xmlns="http://www.w3.org/2000/svg" xmlns:wrs="http://www.wiris.com/xml/mathml-extension" height="24" width="81" wrs:baseline="19"><!--MathML: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mn>3</mn><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></msup></math>--><defs><style type="text/css">@font-face{font-family:'aec8956637a99787bd197eacd77acce';src:url(data:font/truetype;charset=utf-8;base64,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)format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'math143f4d31b04031e49f5eb18baba';src:url(data:font/truetype;charset=utf-8;base64,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)format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'round_brackets18549f92a457f2409';src:url(data:font/truetype;charset=utf-8;base64,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)format('truetype');font-weight:normal;font-style:normal;}</style></defs><text font-family="aec8956637a99787bd197eacd77acce" font-size="16" font-style="italic" text-anchor="middle" x="3.5" y="19">f</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="12.5" y="19">(</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="25.5" y="19">)</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="18.5" y="19">x</text><text font-family="math143f4d31b04031e49f5eb18baba" font-size="16" text-anchor="middle" x="36.5" y="19">=</text><text font-family="Arial" font-size="16" text-anchor="middle" x="49.5" y="19">3</text><text font-family="Arial" font-size="12" font-style="italic" text-anchor="middle" x="57.5" y="12">x</text><text font-family="math143f4d31b04031e49f5eb18baba" font-size="12" text-anchor="middle" x="67.5" y="12">&#x2212;</text><text font-family="Arial" font-size="12" text-anchor="middle" x="76.5" y="12">2</text></svg>" role="math" localid="1647080885832" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/c972c1a2-f723-4530-b34a-922dc29c881c.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220312%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220312T104303Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=39715aac1eb26a2eec026043718d304ea3a2611941ce822f8bfc0374dd8b4ec0" f x = 3 x - 2 .

For graphing $f\left(x\right)=3^{x-2}+1$, we shift the graph of $f\left(x\right)=3^{x-2}$vertically up by $1$ unit.

From the graph we can see that the domain of $f\left(x\right)=3^{x-2}+1$ consists of all the real numbers and the range of $f$ is the set of all the real numbers $y$ for which $y>1$ and the graph has a horizontal asymptote at $y=1$.