Math Problem Statement

Solve all of them

Solution

Let's solve the given problems involving inverse and composite functions.

Given functions: f(x)=2x53f(x) = \frac{2x - 5}{3} g(x)=117x6g(x) = 11 - \frac{7x}{6}

We need to find the following:

  1. f1(x)f^{-1}(x)
  2. g1(x)g^{-1}(x)
  3. f(g(x))f(g(x))
  4. g(f(x))g(f(x))
  5. g1(f(x))g^{-1}(f(x))
  6. f(g1(x))f(g^{-1}(x))
  7. g(f1(x))g(f^{-1}(x))

1. Find f1(x)f^{-1}(x)

To find the inverse of f(x)f(x):

y=2x53y = \frac{2x - 5}{3} 3y=2x53y = 2x - 5 2x=3y+52x = 3y + 5 x=3y+52x = \frac{3y + 5}{2}

So,

f1(x)=3x+52f^{-1}(x) = \frac{3x + 5}{2}

2. Find g1(x)g^{-1}(x)

To find the inverse of g(x)g(x):

y=117x6y = 11 - \frac{7x}{6} 7x6=11y\frac{7x}{6} = 11 - y 7x=6(11y)7x = 6(11 - y) 7x=666y7x = 66 - 6y x=666y7x = \frac{66 - 6y}{7}

So,

g1(x)=666x7g^{-1}(x) = \frac{66 - 6x}{7}

3. Find f(g(x))f(g(x))

g(x)=117x6g(x) = 11 - \frac{7x}{6} f(g(x))=f(117x6)f(g(x)) = f\left(11 - \frac{7x}{6}\right) =2(117x6)53= \frac{2\left(11 - \frac{7x}{6}\right) - 5}{3} =2×112×7x653= \frac{2 \times 11 - 2 \times \frac{7x}{6} - 5}{3} =2214x653= \frac{22 - \frac{14x}{6} - 5}{3} =227x353= \frac{22 - \frac{7x}{3} - 5}{3} =177x33= \frac{17 - \frac{7x}{3}}{3} =17×37x3×3= \frac{17 \times 3 - 7x}{3 \times 3} =517x9= \frac{51 - 7x}{9}

4. Find g(f(x))g(f(x))

f(x)=2x53f(x) = \frac{2x - 5}{3} g(f(x))=g(2x53)g(f(x)) = g\left(\frac{2x - 5}{3}\right) =117(2x53)6= 11 - \frac{7\left(\frac{2x - 5}{3}\right)}{6} =117(2x5)18= 11 - \frac{7(2x - 5)}{18} =1114x3518= 11 - \frac{14x - 35}{18} =1114x18+3518= 11 - \frac{14x}{18} + \frac{35}{18} =117x9+3518= 11 - \frac{7x}{9} + \frac{35}{18} =117x9+3518= 11 - \frac{7x}{9} + \frac{35}{18} =1987x+3518= \frac{198 - 7x + 35}{18} =2337x18= \frac{233 - 7x}{18}

5. Find g1(f(x))g^{-1}(f(x))

f(x)=2x53f(x) = \frac{2x - 5}{3} g1(x)=666x7g^{-1}(x) = \frac{66 - 6x}{7} g1(f(x))=g1(2x53)g^{-1}(f(x)) = g^{-1}\left(\frac{2x - 5}{3}\right) =666(2x53)7= \frac{66 - 6\left(\frac{2x - 5}{3}\right)}{7} =6612x3037= \frac{66 - \frac{12x - 30}{3}}{7} =664x+107= \frac{66 - 4x + 10}{7} =764x7= \frac{76 - 4x}{7}

6. Find f(g1(x))f(g^{-1}(x))

g1(x)=666x7g^{-1}(x) = \frac{66 - 6x}{7} f(g1(x))=f(666x7)f(g^{-1}(x)) = f\left(\frac{66 - 6x}{7}\right) =2(666x7)53= \frac{2\left(\frac{66 - 6x}{7}\right) - 5}{3} =13212x753= \frac{\frac{132 - 12x}{7} - 5}{3} =13212x3573= \frac{\frac{132 - 12x - 35}{7}}{3} =9712x73= \frac{\frac{97 - 12x}{7}}{3} =9712x21= \frac{97 - 12x}{21}

7. Find g(f1(x))g(f^{-1}(x))

f1(x)=3x+52f^{-1}(x) = \frac{3x + 5}{2} g(f1(x))=g(3x+52)g(f^{-1}(x)) = g\left(\frac{3x + 5}{2}\right) =117(3x+52)6= 11 - \frac{7\left(\frac{3x + 5}{2}\right)}{6} =117(3x+5)12= 11 - \frac{7(3x + 5)}{12} =1121x+3512= 11 - \frac{21x + 35}{12} =1121x123512= 11 - \frac{21x}{12} - \frac{35}{12} =13221x3512= \frac{132 - 21x - 35}{12} =9721x12= \frac{97 - 21x}{12}

Do you have any questions or need further details on any of these solutions?

Here are 5 additional questions you might ask:

  1. How do you verify if two functions are inverses of each other?
  2. Can you explain the concept of composite functions in more detail?
  3. What are some common properties of inverse functions?
  4. How can you determine if a function has an inverse?
  5. What are some real-world applications of inverse functions?

Tip:

When finding the inverse of a function, always ensure to swap the dependent and independent variables and then solve for the new dependent variable. This helps in reversing the effect of the original function.

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Math Problem Analysis

Mathematical Concepts

Functions
Inverse Functions
Composite Functions

Formulas

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Theorems

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Suitable Grade Level

Grades 11-12