Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Compotition of two finctions: Adrenced
For the real-yalued functions 
f(x)=sqrt(3x+30) and 
g(x)=x-5, find the composition 
f@g and specfy its domain using interyel notation

(f@z)(x)=◻
Domain of 
f@g:





()/()
口
^(P)

sqrt◻


미미

(0,,◻)

[0,◻]


미

(0,◻]

[0,◻)



O/

oo

-oo



x

3

Compotition of two finctions: Adrenced\newlineFor the real-yalued functions f(x)=3x+30 f(x)=\sqrt{3 x+30} and g(x)=x5 g(x)=x-5 , find the composition fg f \circ g and specfy its domain using interyel notation\newline(fz)(x)= (f \circ z)(x)=\square \newlineDomain of fg: f \circ g: \newline\begin{tabular}{|ccc}\newline\hline \frac{}{} & 口P ^{P} & \sqrt{\square} \\\newline미미 & (0,,) (0,, \square) & {[0,] [0, \square] } \\\newline미 & (0,] (0, \square] & {g(x)=x5 g(x)=x-5 00} \\\newlineg(x)=x5 g(x)=x-5 11 & g(x)=x5 g(x)=x-5 22 & g(x)=x5 g(x)=x-5 33 \\\newlineg(x)=x5 g(x)=x-5 44 & & 33 \\\newline\hline\newline\end{tabular}

Full solution

Q. Compotition of two finctions: Adrenced\newlineFor the real-yalued functions f(x)=3x+30 f(x)=\sqrt{3 x+30} and g(x)=x5 g(x)=x-5 , find the composition fg f \circ g and specfy its domain using interyel notation\newline(fz)(x)= (f \circ z)(x)=\square \newlineDomain of fg: f \circ g: \newline\begin{tabular}{|ccc}\newline\hline \frac{}{} & 口P ^{P} & \sqrt{\square} \\\newline미미 & (0,,) (0,, \square) & {[0,] [0, \square] } \\\newline미 & (0,] (0, \square] & {g(x)=x5 g(x)=x-5 00} \\\newlineg(x)=x5 g(x)=x-5 11 & g(x)=x5 g(x)=x-5 22 & g(x)=x5 g(x)=x-5 33 \\\newlineg(x)=x5 g(x)=x-5 44 & & 33 \\\newline\hline\newline\end{tabular}
  1. Define f(g(x))f(g(x)): Define the composition f(g(x))f(g(x)).\newlinef(g(x))=f(x5)=3(x5)+30f(g(x)) = f(x - 5) = \sqrt{3(x - 5) + 30}.\newlineCalculation: 3(x5)+30=3x15+30=3x+153(x - 5) + 30 = 3x - 15 + 30 = 3x + 15.
  2. Simplify f(g(x))f(g(x)): Simplify f(g(x))f(g(x)).\newlinef(g(x))=3x+15f(g(x)) = \sqrt{3x + 15}.
  3. Determine domain: Determine the domain of f(g(x))f(g(x)).\newlineFor 3x+15\sqrt{3x + 15} to be defined, 3x+1503x + 15 \geq 0.\newlineCalculation: 3x+1503x15x53x + 15 \geq 0 \rightarrow 3x \geq -15 \rightarrow x \geq -5.
  4. Write in interval notation: Write the domain in interval notation.\newlineDomain of f(g(x))=[5,)f(g(x)) = [-5, \infty).

More problems from Compare linear and exponential growth