Respuesta :

gmany

Answer:

a) 500

Step-by-step explanation:

[tex](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[/tex]

The value of f(g(2)) is 500 and this can be determined by using the arithmetic operations and also by using the logarithmic properties.

Given :

  • [tex]\rm f(x) = 5e^{3x}[/tex]
  • [tex]\rm g(x) = \dfrac{2}{3}ln(5x)[/tex]

The following calculation can be used in order to determine the value of f(g(2)).

First, determine the value of g(2).

[tex]\rm g(2) = \dfrac{2}{3}ln(10)[/tex]

Now, the expression of f(g(x)) is given by:

[tex]\rm f(g(x)) = 5e^{3\times \frac{2}{3}ln(5x)}[/tex]

Now, substitute the (x = 2) in the above expression.

[tex]\rm f(g(2)) = 5e^{2ln(10)}[/tex]

Apply the logarithmic properties in the above expression.

[tex]\rm f(g(2)) = 5e^{ln(10)^{2}}[/tex]

The above expression is reduced to:

[tex]\rm f(g(2)) = 5\times (10)^{{2}}[/tex]

f(g(2)) = 500

For more information, refer to the link given below:

https://brainly.com/question/13101306

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