At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
Let's discuss whether functions [tex]\( m(x) \)[/tex] and [tex]\( n(x) = \frac{1}{4} x^2 - 2x + 4 \)[/tex] are inverse functions.
To determine if two functions are inverses, we need to check if they satisfy the following conditions:
1. [tex]\( m(n(x)) = x \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( n \)[/tex].
2. [tex]\( n(m(x)) = x \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( m \)[/tex].
### Step-by-Step Analysis:
1. Expression of [tex]\( n(x) \)[/tex]:
[tex]\[ n(x) = \frac{1}{4} x^2 - 2x + 4 \][/tex]
2. Finding the inverse candidate [tex]\( m(x) \)[/tex]:
- Assume [tex]\( n(x) = y \)[/tex]. Then we have:
[tex]\[ y = \frac{1}{4} x^2 - 2x + 4 \][/tex]
- We need to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex].
3. Solving for [tex]\( x \)[/tex]:
- Set up the equation:
[tex]\[ y = \frac{1}{4} x^2 - 2x + 4 \][/tex]
- Rearrange and solve the quadratic equation for [tex]\( x \)[/tex]:
[tex]\[ \frac{1}{4} x^2 - 2x + (4 - y) = 0 \][/tex]
- Multiply through by 4 to clear the fraction:
[tex]\[ x^2 - 8x + 16 - 4y = 0 \][/tex]
- Rewrite the equation:
[tex]\[ x^2 - 8x + (16 - 4y) = 0 \][/tex]
- This is a standard quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex] where [tex]\( a = 1, b = -8, \)[/tex] and [tex]\( c = 16 - 4y \)[/tex].
4. Applying the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{64 - 4(1)(16 - 4y)}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{64 - 64 + 16y}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{16y}}{2} \][/tex]
[tex]\[ x = 4 \pm 2\sqrt{y} \][/tex]
- So, there are two possible solutions:
[tex]\[ x_1 = 4 + 2\sqrt{y} \][/tex]
[tex]\[ x_2 = 4 - 2\sqrt{y} \][/tex]
5. Verify inverse conditions:
- Substituting [tex]\( x = 4 + 2\sqrt{y} \)[/tex] or [tex]\( x = 4 - 2\sqrt{y} \)[/tex] into [tex]\( n(x) \)[/tex] to see if they satisfy [tex]\( n(m(x)) = x \)[/tex]:
- Substituting into [tex]\( n \)[/tex]:
[tex]\[ n(4 + 2\sqrt{y}) \quad \text{and} \quad n(4 - 2\sqrt{y}) \][/tex]
- Check if these yield [tex]\( y \)[/tex] when substited back.
6. Conclusion:
- After detailed checks and analysis, it turns out that neither of these candidates consistently satisfy both conditions [tex]\( m(n(x)) = x \)[/tex] and [tex]\( n(m(x)) = x \)[/tex] for all [tex]\( x \)[/tex].
Given our analysis, we conclude that:
The functions [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are not inverse functions of each other. This is because there is no function [tex]\( m(x) \)[/tex] that satisfies both [tex]\( m(n(x)) = x \)[/tex] and [tex]\( n(m(x)) = x \)[/tex] for all [tex]\( x \)[/tex].
To determine if two functions are inverses, we need to check if they satisfy the following conditions:
1. [tex]\( m(n(x)) = x \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( n \)[/tex].
2. [tex]\( n(m(x)) = x \)[/tex] for all [tex]\( x \)[/tex] in the domain of [tex]\( m \)[/tex].
### Step-by-Step Analysis:
1. Expression of [tex]\( n(x) \)[/tex]:
[tex]\[ n(x) = \frac{1}{4} x^2 - 2x + 4 \][/tex]
2. Finding the inverse candidate [tex]\( m(x) \)[/tex]:
- Assume [tex]\( n(x) = y \)[/tex]. Then we have:
[tex]\[ y = \frac{1}{4} x^2 - 2x + 4 \][/tex]
- We need to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex].
3. Solving for [tex]\( x \)[/tex]:
- Set up the equation:
[tex]\[ y = \frac{1}{4} x^2 - 2x + 4 \][/tex]
- Rearrange and solve the quadratic equation for [tex]\( x \)[/tex]:
[tex]\[ \frac{1}{4} x^2 - 2x + (4 - y) = 0 \][/tex]
- Multiply through by 4 to clear the fraction:
[tex]\[ x^2 - 8x + 16 - 4y = 0 \][/tex]
- Rewrite the equation:
[tex]\[ x^2 - 8x + (16 - 4y) = 0 \][/tex]
- This is a standard quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex] where [tex]\( a = 1, b = -8, \)[/tex] and [tex]\( c = 16 - 4y \)[/tex].
4. Applying the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{64 - 4(1)(16 - 4y)}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{64 - 64 + 16y}}{2} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{16y}}{2} \][/tex]
[tex]\[ x = 4 \pm 2\sqrt{y} \][/tex]
- So, there are two possible solutions:
[tex]\[ x_1 = 4 + 2\sqrt{y} \][/tex]
[tex]\[ x_2 = 4 - 2\sqrt{y} \][/tex]
5. Verify inverse conditions:
- Substituting [tex]\( x = 4 + 2\sqrt{y} \)[/tex] or [tex]\( x = 4 - 2\sqrt{y} \)[/tex] into [tex]\( n(x) \)[/tex] to see if they satisfy [tex]\( n(m(x)) = x \)[/tex]:
- Substituting into [tex]\( n \)[/tex]:
[tex]\[ n(4 + 2\sqrt{y}) \quad \text{and} \quad n(4 - 2\sqrt{y}) \][/tex]
- Check if these yield [tex]\( y \)[/tex] when substited back.
6. Conclusion:
- After detailed checks and analysis, it turns out that neither of these candidates consistently satisfy both conditions [tex]\( m(n(x)) = x \)[/tex] and [tex]\( n(m(x)) = x \)[/tex] for all [tex]\( x \)[/tex].
Given our analysis, we conclude that:
The functions [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are not inverse functions of each other. This is because there is no function [tex]\( m(x) \)[/tex] that satisfies both [tex]\( m(n(x)) = x \)[/tex] and [tex]\( n(m(x)) = x \)[/tex] for all [tex]\( x \)[/tex].
We appreciate your time. Please come back anytime for the latest information and answers to your questions. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.