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={(32,1),(53,4),(94,7),(116,10)}$$g^{-1}$$={\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={(1,32),(4,53),(7,94),(10,116)}g={\left\lbrace(1,\frac32),(4,\frac53),(7,\frac94),(10,\frac{11}{6})\right\rbrace}

g−1={(32,1),(53,4),(94,7),(116,10)}$$g^{-1}$$={\left\lbrace(\frac32,1),(\frac53,4),(\frac94,7),(\frac{11}{6},10)\right\rbrace}

g−1={(23,1),(35,14),(49,17),(611,110)}$$g^{-1}$$={\left\lbrace(\frac23,1),\left(\frac35,\frac14\right),(\frac49,\frac17),(\frac{6}{11},\frac{1}{10})\right\rbrace}

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1. Review of Real Numbers2h 43m

Simplifying Fractions
30m

Evaluating Exponents
29m

Evaluating Expressions
15m

Translating Phrases to Expressions
16m

Translating Sentences to Equations
9m

The Distributive Property
17m

Simplifying Expressions
44m

2. Linear Equations and Inequalities5h 35m

Solving Linear Equations
2h 23m

Linear Inequalities in One Variable
46m

Set Operations and Compound Inequalities
1h 7m

Absolute Value Equations
38m

Absolute Value Inequalities
39m

3. Solving Word Problems2h 46m

Introduction to Problem Solving
37m

Formulas
25m

Percent Problem Solving
59m

Mixture Problem Solving
43m

4. Graphs and Functions5h 12m

The Rectangular Coordinate System
44m

Graph Linear Equations in Two Variables
24m

Graph Linear Equations Using Intercepts
23m

Slope of a Line
44m

Slope-Intercept Form
38m

Point Slope Form
22m

Linear Inequalities in Two Variables
28m

Introduction to Relations and Functions
53m

Function Notation
15m

Composition of Functions
17m

5. Systems of Linear Equations1h 53m

Solving Systems of Linear Equations by Graphing
29m

Solving Systems of Linear Equations by Substitution
15m

Solving Systems of Linear Equations by Elimination
45m

Systems of Linear Inequalities
23m

6. Exponents, Polynomials, and Polynomial Functions3h 17m

The Product Rule
9m

The Quotient Rule
10m

Intro to the Power Rules
17m

The Power of a Quotient Rule
13m

Negative Exponents
24m

Simplifying Exponential Expressions Using All Exponent Rules
6m

Intro to Polynomials
21m

Adding and Subtracting Polynomials
20m

Multiplying Polynomials
38m

Special Products
34m

7. Factoring2h 49m

Factoring by Greatest Common Factor & Grouping
38m

Factoring Trinomials of the Form x² + bx + c
11m

Factoring Trinomials of the Form ax² + bx + c
37m

Factoring Special Products
41m

Quadratic Equations & Applications
41m

8. Rational Expressions and Functions3h 44m

Simplifying Rational Expressions
42m

Multiplying and Dividing Rational Expressions
25m

Adding and Subtracting Rational Expressions with Common Denominators
19m

Least Common Denominators
32m

Adding and Subtracting Rational Expressions with Different Denominators
32m

Rational Equations
44m

Direct & Inverse Variation
27m

9. Roots, Radicals, and Complex Numbers3h 57m

Introduction to Radical Expressions
47m

Simplifying Radical Expressions
47m

Adding and Subtracting Radicals
19m

Multiplying, Dividing, and Rationalizing Radicals
53m

Introduction to Complex Numbers
18m

Multiplying and Dividing Complex Numbers
51m

10. Quadratic Equations and Functions3h 1m

The Square Root Property
25m

Completing the Square
26m

The Quadratic Formula
40m

Graphing Quadratic Equations
1h 28m

11. Inverse, Exponential, & Logarithmic Functions2h 30m

Introduction to Inverse Functions
25m

Exponential Functions
35m

Introduction to Logarithmic Functions
33m

Properties of Logarithms
54m

12. Conic Sections & Systems of Nonlinear Equations2h 24m

Graphing Circles
32m

Ellipses
35m

Parabolas
35m

Hyperbolas
41m

13. Sequences, Series, and the Binomial Theorem1h 50m

Introduction to Sequences
40m

Arithmetic Sequences
27m

Geometric Sequences
29m

Factorials
13m

11. Inverse, Exponential, & Logarithmic Functions

Introduction to Inverse Functions

Intermediate Algebra

11. Inverse, Exponential, & Logarithmic Functions

Introduction to Inverse Functions

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

$g^{-1}={$\left\[\lbrace$(1,$\frac$23),$\left$($\frac$14,$\frac$35$\right$),($\frac$17,$\frac$49),($\frac{1}{10}$,$\frac{6}{11}$)$\right\]\rbrace$}$

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

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

The function is not one-to-one.

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Problema precedenteProssimo problema

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Identify the function for which you want to find the inverse. Let's denote the original function as $f(x)$.

Replace $f(x)$ with $y$ to make the equation easier to work with: write $y = f(x)$.

Swap the variables $x$ and $y$ in the equation. This means you rewrite the equation as $x = f(y)$, which is the key step in finding the inverse function.

Solve the new equation for $y$ in terms of $x$. This involves algebraic manipulation such as isolating $y$ on one side of the equation.

Once you have $y$ expressed as a function of $x$, rewrite it as $f^{-1}(x)$, which represents the inverse function.