Question

Let f(x)=5e^3x and g(x)=2/3ln(5x); find (f(g))(2)

Answer 1

Answer:

a) 500

Step-by-step explanation:

$(f\circ g)(x)=f\bigg(g(x)\bigg)\\\\(f\circ g)(2)=f\bigg(g(2)\bigg)\\\\f(x)=5e^{3x},\ g(x)=\dfrac{2}{3}\ln(5x)\\\\(f\circ g)(x)\to\text{in}\ f(x)\ \text{replace}\ x\ \text{with the expression}\ \dfrac{2}{3}\ln(5x):\\\\(f\circ g)(x)=5e^{3\cdot\frac{2}{3}\ln(5x)}=5e^{2\ln(5x)}\qquad\text{use}\ \ln a^n=n\ln a\\\\(f\circ g)(x)=5e^{\ln(5x)^2}\qquad\text{use}\ a^{\log_ab}=b\\\\(f\circ g)(x)=5(5x)^2=5(5^2x^2)=5(25x^2)=125x^2\\\\(f\circ g)(2)\to\text{put}\ x=2\\\\(f\circ g)(2)=125(2^2)=125(4)=500$