Answer :
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].
