Given
[tex]\begin{gathered} f(x)=5x^2+4x \\ g(x)=\sqrt{2x+3} \end{gathered}[/tex]
To find:
[tex]f\circ g\text{ }and\text{ }f+g[/tex]
And, the domain in interval notation.
Explanation:
It is given that,
[tex]\begin{gathered} f(x)=5x^2+4x \\ g(x)=\sqrt{2x+3} \end{gathered}[/tex]
That implies,
[tex]\begin{gathered} (f\circ g)(x)=f(g(x)) \\ =f(\sqrt{2x+3}) \\ =5(\sqrt{2x+3})^2+4(\sqrt{2x+3}) \\ =5(2x+3)+4\sqrt{2x+3} \\ =10x+15+4\sqrt{2x+3} \end{gathered}[/tex]
And its domain is,
[tex]\begin{gathered} x\ge-\frac{3}{2} \\ Domain:[-\frac{3}{2},\infty) \end{gathered}[/tex]
Also,
[tex]\begin{gathered} (f+g)(x)=f(x)+g(x) \\ =(5x^2+4x)+(\sqrt{2x+3}) \\ =5x^2+4x+\sqrt{2x+3} \end{gathered}[/tex]
And its domain is,
[tex]\begin{gathered} x\ge-\frac{3}{2} \\ \text{ }[-\frac{3}{2},\infty) \end{gathered}[/tex]
