Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
To determine the value of [tex]\( x \)[/tex] when [tex]\( g(h(x)) = 4 \)[/tex], let's follow these steps:
1. Identify the given functions:
- [tex]\( h(x) = 2x + 1 \)[/tex]
- [tex]\( g(x) = x^2 - 4 \)[/tex]
2. Find [tex]\( g(h(x)) \)[/tex]:
Substitute [tex]\( h(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(h(x)) = g(2x + 1) \][/tex]
Since [tex]\( g(x) = x^2 - 4 \)[/tex], replace [tex]\( x \)[/tex] with [tex]\( 2x + 1 \)[/tex]:
[tex]\[ g(2x + 1) = (2x + 1)^2 - 4 \][/tex]
3. Simplify [tex]\( g(h(x)) \)[/tex]:
Calculate [tex]\( (2x + 1)^2 \)[/tex]:
[tex]\[ (2x + 1)^2 = 4x^2 + 4x + 1 \][/tex]
Subtract 4:
[tex]\[ g(2x + 1) = 4x^2 + 4x + 1 - 4 = 4x^2 + 4x - 3 \][/tex]
4. Set [tex]\( g(h(x)) = 4 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 4x^2 + 4x - 3 = 4 \][/tex]
Rearrange the equation:
[tex]\[ 4x^2 + 4x - 3 - 4 = 0 \][/tex]
Simplify:
[tex]\[ 4x^2 + 4x - 7 = 0 \][/tex]
5. Solve the quadratic equation [tex]\( 4x^2 + 4x - 7 = 0 \)[/tex]:
To solve the quadratic equation, we can use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 4 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -7 \)[/tex].
Calculate the discriminant:
[tex]\[ b^2 - 4ac = 4^2 - 4 \cdot 4 \cdot (-7) = 16 + 112 = 128 \][/tex]
Since the discriminant is positive, there are two real solutions:
[tex]\[ x = \frac{-4 \pm \sqrt{128}}{8} = \frac{-4 \pm 8\sqrt{2}}{8} = \frac{-1 \pm 2\sqrt{2}}{2} \][/tex]
Thus, the solutions are:
[tex]\[ x_1 = \frac{-1 + 2\sqrt{2}}{2} \quad \text{and} \quad x_2 = \frac{-1 - 2\sqrt{2}}{2} \][/tex]
6. Evaluate the given options [tex]\( 0, 2, 4, 5 \)[/tex]:
None of the given options (0, 2, 4, 5) correspond to the roots [tex]\( \frac{-1 + 2\sqrt{2}}{2} \)[/tex] or [tex]\( \frac{-1 - 2\sqrt{2}}{2} \)[/tex].
Therefore, given the provided choices and our calculations, none of the options (0, 2, 4, 5) are correct. The actual result is that there is no matching [tex]\( x \)[/tex] value among the given options that satisfies the equation [tex]\( g(h(x)) = 4 \)[/tex]. Hence, the answer is:
```
None
```
1. Identify the given functions:
- [tex]\( h(x) = 2x + 1 \)[/tex]
- [tex]\( g(x) = x^2 - 4 \)[/tex]
2. Find [tex]\( g(h(x)) \)[/tex]:
Substitute [tex]\( h(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(h(x)) = g(2x + 1) \][/tex]
Since [tex]\( g(x) = x^2 - 4 \)[/tex], replace [tex]\( x \)[/tex] with [tex]\( 2x + 1 \)[/tex]:
[tex]\[ g(2x + 1) = (2x + 1)^2 - 4 \][/tex]
3. Simplify [tex]\( g(h(x)) \)[/tex]:
Calculate [tex]\( (2x + 1)^2 \)[/tex]:
[tex]\[ (2x + 1)^2 = 4x^2 + 4x + 1 \][/tex]
Subtract 4:
[tex]\[ g(2x + 1) = 4x^2 + 4x + 1 - 4 = 4x^2 + 4x - 3 \][/tex]
4. Set [tex]\( g(h(x)) = 4 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 4x^2 + 4x - 3 = 4 \][/tex]
Rearrange the equation:
[tex]\[ 4x^2 + 4x - 3 - 4 = 0 \][/tex]
Simplify:
[tex]\[ 4x^2 + 4x - 7 = 0 \][/tex]
5. Solve the quadratic equation [tex]\( 4x^2 + 4x - 7 = 0 \)[/tex]:
To solve the quadratic equation, we can use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 4 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -7 \)[/tex].
Calculate the discriminant:
[tex]\[ b^2 - 4ac = 4^2 - 4 \cdot 4 \cdot (-7) = 16 + 112 = 128 \][/tex]
Since the discriminant is positive, there are two real solutions:
[tex]\[ x = \frac{-4 \pm \sqrt{128}}{8} = \frac{-4 \pm 8\sqrt{2}}{8} = \frac{-1 \pm 2\sqrt{2}}{2} \][/tex]
Thus, the solutions are:
[tex]\[ x_1 = \frac{-1 + 2\sqrt{2}}{2} \quad \text{and} \quad x_2 = \frac{-1 - 2\sqrt{2}}{2} \][/tex]
6. Evaluate the given options [tex]\( 0, 2, 4, 5 \)[/tex]:
None of the given options (0, 2, 4, 5) correspond to the roots [tex]\( \frac{-1 + 2\sqrt{2}}{2} \)[/tex] or [tex]\( \frac{-1 - 2\sqrt{2}}{2} \)[/tex].
Therefore, given the provided choices and our calculations, none of the options (0, 2, 4, 5) are correct. The actual result is that there is no matching [tex]\( x \)[/tex] value among the given options that satisfies the equation [tex]\( g(h(x)) = 4 \)[/tex]. Hence, the answer is:
```
None
```
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.