Let's solve the given problems involving inverse and composite functions.
Given functions:
f(x)=32x−5
g(x)=11−67x
We need to find the following:
- f−1(x)
- g−1(x)
- f(g(x))
- g(f(x))
- g−1(f(x))
- f(g−1(x))
- g(f−1(x))
1. Find f−1(x)
To find the inverse of f(x):
y=32x−5
3y=2x−5
2x=3y+5
x=23y+5
So,
f−1(x)=23x+5
2. Find g−1(x)
To find the inverse of g(x):
y=11−67x
67x=11−y
7x=6(11−y)
7x=66−6y
x=766−6y
So,
g−1(x)=766−6x
3. Find f(g(x))
g(x)=11−67x
f(g(x))=f(11−67x)
=32(11−67x)−5
=32×11−2×67x−5
=322−614x−5
=322−37x−5
=317−37x
=3×317×3−7x
=951−7x
4. Find g(f(x))
f(x)=32x−5
g(f(x))=g(32x−5)
=11−67(32x−5)
=11−187(2x−5)
=11−1814x−35
=11−1814x+1835
=11−97x+1835
=11−97x+1835
=18198−7x+35
=18233−7x
5. Find g−1(f(x))
f(x)=32x−5
g−1(x)=766−6x
g−1(f(x))=g−1(32x−5)
=766−6(32x−5)
=766−312x−30
=766−4x+10
=776−4x
6. Find f(g−1(x))
g−1(x)=766−6x
f(g−1(x))=f(766−6x)
=32(766−6x)−5
=37132−12x−5
=37132−12x−35
=3797−12x
=2197−12x
7. Find g(f−1(x))
f−1(x)=23x+5
g(f−1(x))=g(23x+5)
=11−67(23x+5)
=11−127(3x+5)
=11−1221x+35
=11−1221x−1235
=12132−21x−35
=1297−21x
Do you have any questions or need further details on any of these solutions?
Here are 5 additional questions you might ask:
- How do you verify if two functions are inverses of each other?
- Can you explain the concept of composite functions in more detail?
- What are some common properties of inverse functions?
- How can you determine if a function has an inverse?
- 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.