Let's solve each part step-by-step:
### Part (i): Solve [tex]\( g(x) = 10 \)[/tex]
Given the function [tex]\( g(x) = e^{(x+3)} \)[/tex], we need to solve for [tex]\( x \)[/tex] when [tex]\( g(x) = 10 \)[/tex].
1. We start with the equation:
[tex]\[ e^{(x+3)} = 10 \][/tex]
2. To solve for [tex]\( x \)[/tex], take the natural logarithm (ln) of both sides. The natural logarithm is the inverse function of the exponential function. Applying the natural logarithm, we get:
[tex]\[ \ln(e^{(x+3)}) = \ln(10) \][/tex]
3. By the properties of logarithms, [tex]\( \ln(e^{(x+3)}) = x + 3 \)[/tex]:
[tex]\[ x + 3 = \ln(10) \][/tex]
4. Solve for [tex]\( x \)[/tex] by isolating it on one side of the equation:
[tex]\[ x = \ln(10) - 3 \][/tex]
Thus, the solution for [tex]\( x \)[/tex] in the equation [tex]\( g(x) = 10 \)[/tex] is:
[tex]\[ x \approx -0.6974 \][/tex]
### Part (ii): Solve [tex]\( h(x) = \frac{\pi}{4} \)[/tex]
Given the function [tex]\( h(x) = \arcsin \left( \frac{x}{2} \right) \)[/tex], we need to solve for [tex]\( x \)[/tex] when [tex]\( h(x) = \frac{\pi}{4} \)[/tex].
1. We start with the equation:
[tex]\[ \arcsin \left( \frac{x}{2} \right) = \frac{\pi}{4} \][/tex]
2. Recall that the arcsine function [tex]\( \arcsin(y) \)[/tex] gives the angle [tex]\( \theta \)[/tex] such that [tex]\( \sin(\theta) = y \)[/tex]. Therefore:
[tex]\[ \sin\left( \frac{\pi}{4} \right) = \frac{x}{2} \][/tex]
3. The sine of [tex]\( \frac{\pi}{4} \)[/tex] is:
[tex]\[ \sin\left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \][/tex]
4. Substitute [tex]\( \frac{\sqrt{2}}{2} \)[/tex] into the equation:
[tex]\[ \frac{\sqrt{2}}{2} = \frac{x}{2} \][/tex]
5. Solve for [tex]\( x \)[/tex] by multiplying both sides by 2:
[tex]\[ x = 2 \cdot \frac{\sqrt{2}}{2} = \sqrt{2} \][/tex]
Since [tex]\( \sqrt{2} \)[/tex] is approximately 1.4142, we can write:
[tex]\[ x \approx 1.4142 \][/tex]
Thus, the solution for [tex]\( x \)[/tex] in the equation [tex]\( h(x) = \frac{\pi}{4} \)[/tex] is:
[tex]\[ x \approx 1.4142 \][/tex]
### Summary of Solutions
- The solution to [tex]\( g(x) = 10 \)[/tex] is [tex]\( x \approx -0.6974 \)[/tex].
- The solution to [tex]\( h(x) = \frac{\pi}{4} \)[/tex] is [tex]\( x \approx 1.4142 \)[/tex].
