Connect with a community of experts and enthusiasts on IDNLearn.com. Our experts provide timely and precise responses to help you understand and solve any issue you face.
Sagot :
Sure, let's find the inverse functions for each given function and verify them by composition.
### Function [tex]\( f(x) = -2x + 3 \)[/tex]
1. Find the inverse function:
To find the inverse function, we start by replacing [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = -2x + 3 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ y - 3 = -2x \implies x = \frac{3 - y}{2} \][/tex]
Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to get the inverse function:
[tex]\[ f^{-1}(x) = \frac{3 - x}{2} \][/tex]
2. Verify by composition:
To verify that [tex]\( f^{-1}(x) \)[/tex] is indeed the inverse of [tex]\( f(x) \)[/tex], we need to check that:
[tex]\[ f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x \][/tex]
- Verify [tex]\( f(f^{-1}(x)) \)[/tex]:
[tex]\[ f\left( f^{-1}(x) \right) = f\left( \frac{3 - x}{2} \right) = -2 \left( \frac{3 - x}{2} \right) + 3 = - (3 - x) + 3 = x \][/tex]
- Verify [tex]\( f^{-1}(f(x)) \)[/tex]:
[tex]\[ f^{-1}(f(x)) = f^{-1}(-2x + 3) = \frac{3 - (-2x + 3)}{2} = \frac{3 + 2x - 3}{2} = x \][/tex]
3. Graph:
To graph both [tex]\( f(x) \)[/tex] and [tex]\( f^{-1}(x) \)[/tex], plot the lines:
[tex]\[ f(x) = -2x + 3 \quad \text{and} \quad f^{-1}(x) = \frac{3 - x}{2} \][/tex]
The line [tex]\( y = x \)[/tex] will also be plotted as a reference as it represents the reflection line for the inverse functions.
### Function [tex]\( f(x) = 2x - 3 \)[/tex]
1. Find the inverse function:
Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = 2x - 3 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ y + 3 = 2x \implies x = \frac{y + 3}{2} \][/tex]
Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to get the inverse function:
[tex]\[ f^{-1}(x) = \frac{x + 3}{2} \][/tex]
2. Verify by composition:
To verify that [tex]\( f^{-1}(x) \)[/tex] is indeed the inverse of [tex]\( f(x) \)[/tex], we need to check that:
[tex]\[ f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x \][/tex]
- Verify [tex]\( f(f^{-1}(x)) \)[/tex]:
[tex]\[ f\left( f^{-1}(x) \right) = f\left( \frac{x + 3}{2} \right) = 2 \left( \frac{x + 3}{2} \right) - 3 = x + 3 - 3 = x \][/tex]
- Verify [tex]\( f^{-1}(f(x)) \)[/tex]:
[tex]\[ f^{-1}(f(x)) = f^{-1}(2x - 3) = \frac{(2x - 3) + 3}{2} = \frac{2x}{2} = x \][/tex]
3. Graph:
To graph both [tex]\( f(x) \)[/tex] and [tex]\( f^{-1}(x) \)[/tex], plot the lines:
[tex]\[ f(x) = 2x - 3 \quad \text{and} \quad f^{-1}(x) = \frac{x + 3}{2} \][/tex]
Again, the line [tex]\( y = x \)[/tex] will be plotted as a reference.
These steps provide clear guidance on finding the inverse functions for [tex]\( f(x) = -2x + 3 \)[/tex] and [tex]\( f(x) = 2x - 3 \)[/tex], verifying them through composition, and graphing the functions along with their inverses.
### Function [tex]\( f(x) = -2x + 3 \)[/tex]
1. Find the inverse function:
To find the inverse function, we start by replacing [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = -2x + 3 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ y - 3 = -2x \implies x = \frac{3 - y}{2} \][/tex]
Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to get the inverse function:
[tex]\[ f^{-1}(x) = \frac{3 - x}{2} \][/tex]
2. Verify by composition:
To verify that [tex]\( f^{-1}(x) \)[/tex] is indeed the inverse of [tex]\( f(x) \)[/tex], we need to check that:
[tex]\[ f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x \][/tex]
- Verify [tex]\( f(f^{-1}(x)) \)[/tex]:
[tex]\[ f\left( f^{-1}(x) \right) = f\left( \frac{3 - x}{2} \right) = -2 \left( \frac{3 - x}{2} \right) + 3 = - (3 - x) + 3 = x \][/tex]
- Verify [tex]\( f^{-1}(f(x)) \)[/tex]:
[tex]\[ f^{-1}(f(x)) = f^{-1}(-2x + 3) = \frac{3 - (-2x + 3)}{2} = \frac{3 + 2x - 3}{2} = x \][/tex]
3. Graph:
To graph both [tex]\( f(x) \)[/tex] and [tex]\( f^{-1}(x) \)[/tex], plot the lines:
[tex]\[ f(x) = -2x + 3 \quad \text{and} \quad f^{-1}(x) = \frac{3 - x}{2} \][/tex]
The line [tex]\( y = x \)[/tex] will also be plotted as a reference as it represents the reflection line for the inverse functions.
### Function [tex]\( f(x) = 2x - 3 \)[/tex]
1. Find the inverse function:
Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = 2x - 3 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ y + 3 = 2x \implies x = \frac{y + 3}{2} \][/tex]
Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to get the inverse function:
[tex]\[ f^{-1}(x) = \frac{x + 3}{2} \][/tex]
2. Verify by composition:
To verify that [tex]\( f^{-1}(x) \)[/tex] is indeed the inverse of [tex]\( f(x) \)[/tex], we need to check that:
[tex]\[ f(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = x \][/tex]
- Verify [tex]\( f(f^{-1}(x)) \)[/tex]:
[tex]\[ f\left( f^{-1}(x) \right) = f\left( \frac{x + 3}{2} \right) = 2 \left( \frac{x + 3}{2} \right) - 3 = x + 3 - 3 = x \][/tex]
- Verify [tex]\( f^{-1}(f(x)) \)[/tex]:
[tex]\[ f^{-1}(f(x)) = f^{-1}(2x - 3) = \frac{(2x - 3) + 3}{2} = \frac{2x}{2} = x \][/tex]
3. Graph:
To graph both [tex]\( f(x) \)[/tex] and [tex]\( f^{-1}(x) \)[/tex], plot the lines:
[tex]\[ f(x) = 2x - 3 \quad \text{and} \quad f^{-1}(x) = \frac{x + 3}{2} \][/tex]
Again, the line [tex]\( y = x \)[/tex] will be plotted as a reference.
These steps provide clear guidance on finding the inverse functions for [tex]\( f(x) = -2x + 3 \)[/tex] and [tex]\( f(x) = 2x - 3 \)[/tex], verifying them through composition, and graphing the functions along with their inverses.
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. For clear and precise answers, choose IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.
