Answer :
Certainly! Let's go through the steps and identify the errors in finding the inverse of the function [tex]\( y = x^2 + 12x \)[/tex].
### Step-by-Step Analysis:
1. Starting Point:
The given equation is [tex]\( y = x^2 + 12x \)[/tex].
- Goal: To find the inverse, we need to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex].
2. Incorrect Setup:
The work presents the next step as [tex]\( x = y^2 + 12x \)[/tex].
- Error 1: This is incorrect because it doesn't follow logically from the original equation. We should have been solving for [tex]\( x \)[/tex] from the equation [tex]\( y = x^2 + 12x \)[/tex], not introducing [tex]\( y^2 \)[/tex] inappropriately.
3. Solving for [tex]\( x \)[/tex]:
The correct approach is:
[tex]\[ y = x^2 + 12x \implies x^2 + 12x - y = 0 \][/tex]
This is a quadratic equation in terms of [tex]\( x \)[/tex].
4. Applying the Quadratic Formula Correctly:
For a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], the solutions for [tex]\( x \)[/tex] are given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 12 \)[/tex], and [tex]\( c = -y \)[/tex]. Substituting these values:
[tex]\[ x = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 1 \cdot (-y)}}{2 \cdot 1} \implies x = \frac{-12 \pm \sqrt{144 + 4y}}{2} \implies x = \frac{-12 \pm \sqrt{144 + 4y}}{2} \][/tex]
Thus, the correct expression for [tex]\( x \)[/tex] involves using the quadratic formula with the correct coefficients.
5. Improper Conclusion:
The work incorrectly states:
[tex]\[ y^2 = x - 12x \implies y^2 = -11x \implies y = \sqrt{-11x}, \text{ for } x \geq 0 \][/tex]
- Error 2: Correctly solving the quadratic equation should use the quadratic formula rather than simplify directly incorrectly.
- Error 3: Concluding that [tex]\( y = \sqrt{-11x} \)[/tex] is invalid because it involves taking the square root of a negative number when [tex]\( x \geq 0 \)[/tex], which is not within the domain of real numbers.
### Summary of Errors:
1. Incorrect Setup:
- Mistakenly claiming [tex]\( x = y^2 + 12x \)[/tex] from the original equation is incorrect.
2. Incorrect Application of Quadratic Solution:
- Solving the quadratic equation [tex]\( y = x^2 + 12x \)[/tex] for [tex]\( x \)[/tex] should utilize the quadratic formula accurately.
3. Invalid Mathematical Conclusion:
- An improper conclusion [tex]\( y = \sqrt{-11x} \)[/tex] suggesting an invalid domain for [tex]\( x \)[/tex], since the square root of a negative number is not defined within the realm of real numbers for [tex]\( x \geq 0 \)[/tex].
These errors highlight the need for careful algebraic manipulation and correct application of mathematical principles when finding the inverse of functions.
### Step-by-Step Analysis:
1. Starting Point:
The given equation is [tex]\( y = x^2 + 12x \)[/tex].
- Goal: To find the inverse, we need to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex].
2. Incorrect Setup:
The work presents the next step as [tex]\( x = y^2 + 12x \)[/tex].
- Error 1: This is incorrect because it doesn't follow logically from the original equation. We should have been solving for [tex]\( x \)[/tex] from the equation [tex]\( y = x^2 + 12x \)[/tex], not introducing [tex]\( y^2 \)[/tex] inappropriately.
3. Solving for [tex]\( x \)[/tex]:
The correct approach is:
[tex]\[ y = x^2 + 12x \implies x^2 + 12x - y = 0 \][/tex]
This is a quadratic equation in terms of [tex]\( x \)[/tex].
4. Applying the Quadratic Formula Correctly:
For a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], the solutions for [tex]\( x \)[/tex] are given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 12 \)[/tex], and [tex]\( c = -y \)[/tex]. Substituting these values:
[tex]\[ x = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 1 \cdot (-y)}}{2 \cdot 1} \implies x = \frac{-12 \pm \sqrt{144 + 4y}}{2} \implies x = \frac{-12 \pm \sqrt{144 + 4y}}{2} \][/tex]
Thus, the correct expression for [tex]\( x \)[/tex] involves using the quadratic formula with the correct coefficients.
5. Improper Conclusion:
The work incorrectly states:
[tex]\[ y^2 = x - 12x \implies y^2 = -11x \implies y = \sqrt{-11x}, \text{ for } x \geq 0 \][/tex]
- Error 2: Correctly solving the quadratic equation should use the quadratic formula rather than simplify directly incorrectly.
- Error 3: Concluding that [tex]\( y = \sqrt{-11x} \)[/tex] is invalid because it involves taking the square root of a negative number when [tex]\( x \geq 0 \)[/tex], which is not within the domain of real numbers.
### Summary of Errors:
1. Incorrect Setup:
- Mistakenly claiming [tex]\( x = y^2 + 12x \)[/tex] from the original equation is incorrect.
2. Incorrect Application of Quadratic Solution:
- Solving the quadratic equation [tex]\( y = x^2 + 12x \)[/tex] for [tex]\( x \)[/tex] should utilize the quadratic formula accurately.
3. Invalid Mathematical Conclusion:
- An improper conclusion [tex]\( y = \sqrt{-11x} \)[/tex] suggesting an invalid domain for [tex]\( x \)[/tex], since the square root of a negative number is not defined within the realm of real numbers for [tex]\( x \geq 0 \)[/tex].
These errors highlight the need for careful algebraic manipulation and correct application of mathematical principles when finding the inverse of functions.
