The functions 𝑔 and ℎ are given by          𝑔⁡(𝑥)=log5⁡(4⁢𝑥-2)          ℎ⁡(𝑥)=sin-1⁡(8⁢𝑥).(i) Solve 𝑔⁡(𝑥)=3 for values of 𝑥 in the domain of 𝑔.(ii) Solve  ℎ⁡(𝑥)=𝜋4 for values of 𝑥 in the domain of ℎ.

(i) To solve 𝑔(𝑥)=3 for values of 𝑥 in the domain of 𝑔, we first set 𝑔(𝑥) equal to 3:

log5(4𝑥-2) = 3

Next, we convert the logarithmic equation to an exponential equation:

5^3 = 4𝑥 - 2

125 = 4𝑥 - 2

Then, we solve for 𝑥:

4𝑥 = 127

𝑥 = 127/4

𝑥 = 31.75

So, the solution to 𝑔(𝑥)=3 in the domain of 𝑔 is 𝑥 = 31.75.

(ii) To solve ℎ(𝑥)=𝜋/4 for values of 𝑥 in the domain of ℎ, we first set ℎ(𝑥) equal to 𝜋/4:

sin^-1(8𝑥) = 𝜋/4

Next, we take the sine of both sides:

8𝑥 = sin(𝜋/4)

Then, we solve for 𝑥:

𝑥 = sin(𝜋/4) / 8

𝑥 = √2 / 16

So, the solution to ℎ(𝑥)=𝜋/4 in the domain of ℎ is 𝑥 = √2 / 16.

(i) To solve the equation g(x) = 3, we first substitute g(x) with its given function:

log5(4x - 2) = 3

We then convert the logarithmic equation to an exponential equation:

5^3 = 4x - 2

Solving for x, we get:

125 = 4x - 2

Adding 2 to both sides, we get:

127 = 4x

Finally, dividing both sides by 4, we get:

x = 127/4

So, the solution to g(x) = 3 is x = 127/4, provided that x is in the domain of g. The domain of g is x > 1/2, and 127/4 > 1/2, so x = 127/4 is indeed in the domain of g.

(ii) To solve the equation h(x) = π/4, we first substitute h(x) with its given function:

sin^-1(8x) = π/4

We then take the sine of both sides:

8x = sin(π/4)

Solving for x, we get:

x = sin(π/4) / 8

So, the solution to h(x) = π/4 is x = sin(π/4) / 8, provided that x is in the domain of h. The domain of h is -1 ≤ x ≤ 1, and -1 ≤ sin(π/4) / 8 ≤ 1, so x = sin(π/4) / 8 is indeed in the domain of h.