At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Get quick and reliable solutions to your questions from knowledgeable professionals on our comprehensive Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
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
```
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
