A function g is such that $\mathrm{g}(x)=5 x-7$, for $x \in \mathbb{R}$. (ii) Find the exact solution of the equation $\mathrm{gf}(x)=13$.
Explanation
To find the exact solution of the equation gf(x) = 13, we first need to find the composite function gf(x). We have g(x) = 5x - 7 and f(x) = ln(2x + 1). We get gf(x) = g(f(x)) = 5 ln(2x + 1) - 7. Then, we set gf(x) = 13 and solve for x. We get 5 ln(2x + 1) - 7 = 13. We add 7 to both sides to get 5 ln(2x + 1) = 20. We divide both sides by 5 to get ln(2x + 1) = 4. We exponentiate both sides to get 2x + 1 = e^4. We subtract 1 from both sides to get 2x = e^4 - 1. We divide both sides by 2 to get x = (e^4 - 1)/2. However, this is not the only solution. We also need to check the original equation gf(x) = 13. We substitute x = 6 into the equation gf(x) = 5 ln(2x + 1) - 7 to get gf(6) = 5 ln(2*6 + 1) - 7 = 5 ln(13) - 7. We substitute this expression into the equation gf(x) = 13 to get 5 ln(13) - 7 = 13. We add 7 to both sides to get 5 ln(13) = 20. We divide both sides by 5 to get ln(13) = 4. This is a contradiction, since ln(13) is not equal to 4. Therefore, the only solution to the equation gf(x) = 13 is x = 6.
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