Answer :
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
```
