Consider the set of ordered pairs. Verify if it is one-to-one. If so, find its inverse.g={(1,32),(4,53),(7,94),(10,116)}g={\left\lbrace(1,\frac32),(4,\frac53),(7,\frac94),(10,\frac{11}{6})\right\rbrace}

g − 1 = { ( 3 2 , 1 ) , ( 5 3 , 4 ) , ( 9 4 , 7 ) , ( 11 6 , 10 ) } g^{-1}={\left\lbrace(\frac32,1),(\frac53,4),(\frac94,7),(\frac{11}{6},10)\right\rbrace}

Consider the set of ordered pairs. Verify if it is one-to-one. If so, find its inverse. g = { ( 1 , 3 2 ) , ( 4 , 5 3 ) , ( 7 , 9 4 ) , ( 10 , 11 6 ) } g={\left\lbrace(1,\frac32),(4,\frac53),(7,\frac94),(10,\frac{11}{6})\right\rbrace}

we're asked to consider the set of ordered pairs, verify if it is one to one, and if so, find its inverse. So here I have that G is equal to the set of ordered pairs one, three halves, four, five thirds, seven nine fourths, and ten eleven sixths. So before I find the inverse, I have to check that this even has an inverse by verifying that it's one to one. Remember when we're faced with a list of ordered pairs, what we're looking for is either repeated inputs or repeated outputs. Repeated inputs tells me that I'm not dealing with a function. And repeated outputs tells me that even if it is a function, it's not a one to one function. So that's what I'm looking for here. Now here I have no repeated inputs or outputs. I have one input and one output. It is a one to one function. So that means that it does have an inverse and I can proceed in finding it. So my in verse function of g here, that's g with a little negative one up top, this is going to be equal to the set of ordered pairs with reversed x and y values. So one and three one comma three halves, swapping those values. This is three halves, one. And then for my next ordered pair, four, five thirds, this is five thirds, four. Then for seven, nine fourths, swapping those values, nine fourths, seven. And finally, ten, eleven, sixths, swapping those values, I get eleven, six, 10. And then these are the ordered pairs that represent the inverse of g. Let us know if you have any questions, and I'll see you soon.

Consider the set of ordered pairs. Verify if it is one-to-one. If so, find its inverse. g = { ( 1 , 3 2 ) , ( 4 , 5 3 ) , ( 7 , 9 4 ) , ( 10 , 11 6 ) } g={\left\lbrace(1,\frac32),(4,\frac53),(7,\frac94),(10,\frac{11}{6})\right\rbrace}

g − 1 = { ( 3 2 , 1 ) , ( 5 3 , 4 ) , ( 9 4 , 7 ) , ( 11 6 , 10 ) } g^{-1}={\left\lbrace(\frac32,1),(\frac53,4),(\frac94,7),(\frac{11}{6},10)\right\rbrace}

g − 1 = { ( 1 , 2 3 ) , ( 1 4 , 3 5 ) , ( 1 7 , 4 9 ) , ( 1 10 , 6 11 ) } g^{-1}={\left\lbrace(1,\frac23),\left(\frac14,\frac35\right),(\frac17,\frac49),(\frac{1}{10},\frac{6}{11})\right\rbrace}

g − 1 = { ( 2 3 , 1 ) , ( 3 5 , 1 4 ) , ( 4 9 , 1 7 ) , ( 6 11 , 1 10 ) } g^{-1}={\left\lbrace(\frac23,1),\left(\frac35,\frac14\right),(\frac49,\frac17),(\frac{6}{11},\frac{1}{10})\right\rbrace}

13. Inverse, Exponential, & Logarithmic Functions

Introduction to Inverse Functions

Beginning & Intermediate Algebra

13. Inverse, Exponential, & Logarithmic Functions

Introduction to Inverse Functions

Struggling with Beginning & Intermediate Algebra?

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Multiple Choice

Consider the set of ordered pairs. Verify if it is one-to-one. If so, find its inverse.
$g = {(1, \frac{3}{2}), (4, \frac{5}{3}), (7, \frac{9}{4}), (10, \frac{11}{6})}$

$g^{- 1} = {(1, \frac{2}{3}), (\frac{1}{4}, \frac{3}{5}), (\frac{1}{7}, \frac{4}{9}), (\frac{1}{10}, \frac{6}{11})}$

$g^{- 1} = {(\frac{2}{3}, 1), (\frac{3}{5}, \frac{1}{4}), (\frac{4}{9}, \frac{1}{7}), (\frac{6}{11}, \frac{1}{10})}$

$g^{- 1} = {(\frac{3}{2}, 1), (\frac{5}{3}, 4), (\frac{9}{4}, 7), (\frac{11}{6}, 10)}$

The function is not one-to-one.

Verified step by step guidance

Step 1: Understand what it means for a function to be one-to-one (injective). A function is one-to-one if each output value corresponds to exactly one input value. In other words, no two different inputs share the same output.

Step 2: Examine the given set of ordered pairs $g = \left\lbrace (1, \frac{3}{2}), (4, \frac{5}{3}), (7, \frac{9}{4}), (10, \frac{11}{6}) \right\rbrace$. Check if any output values (second elements) repeat for different inputs (first elements).

Step 3: Since all output values $\frac{3}{2}, \frac{5}{3}, \frac{9}{4}, \frac{11}{6}$ are distinct, the function passes the one-to-one test.

Step 4: To find the inverse of the function, swap each ordered pair's components. That is, for each pair $(x, y)$ in $g$, the inverse will have the pair $(y, x)$.

Step 5: Write the inverse set as $g^{-1} = \left\lbrace \left( \frac{3}{2}, 1 \right), \left( \frac{5}{3}, 4 \right), \left( \frac{9}{4}, 7 \right), \left( \frac{11}{6}, 10 \right) \right\rbrace$. This is the inverse relation.

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