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Graphs and Functlons Combining functions: Advanced Suppose that the functions f and g are defined as follows. {:[f(x)=sqrt(x+4)],[g(x)=5x^(2)+2]:} Find f*g and f+g. Then, give their domains using interval notation. (f*g)(x)= Domain of f*g: (f+g)(x)= Domain of f+g :

Graphs and Functlons\newlineCombining functions: Advanced\newlineSuppose that the functions f f and g g are defined as follows.\newlinef(x)=x+4g(x)=5x2+2 \begin{array}{l} f(x)=\sqrt{x+4} \\ g(x)=5 x^{2}+2 \end{array} \newlineFind f⋅g f \cdot g and f+g f+g . Then, give their domains using interval notation.\newline(f⋅g)(x)= (f \cdot g)(x)= \newlineDomain of f⋅g: f \cdot g: \newline(f+g)(x)= (f+g)(x)= \newlineDomain of f+g f+g :

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Q. Graphs and Functlons\newlineCombining functions: Advanced\newlineSuppose that the functions f f and g g are defined as follows.\newlinef(x)=x+4g(x)=5x2+2 \begin{array}{l} f(x)=\sqrt{x+4} \\ g(x)=5 x^{2}+2 \end{array} \newlineFind f⋅g f \cdot g and f+g f+g . Then, give their domains using interval notation.\newline(f⋅g)(x)= (f \cdot g)(x)= \newlineDomain of f⋅g: f \cdot g: \newline(f+g)(x)= (f+g)(x)= \newlineDomain of f+g f+g :
  1. Define Functions and Operations: Define the functions and the operations needed.\newlinef(x)=x+4f(x) = \sqrt{x + 4}, g(x)=5x2+2g(x) = 5x^2 + 2.\newlineWe need to find (f∗g)(x)(f*g)(x) and (f+g)(x)(f+g)(x).
  2. Calculate (f∗g)(x)(f*g)(x): Calculate (f∗g)(x)=f(x)∗g(x)(f*g)(x) = f(x) * g(x).(f∗g)(x)=x+4∗(5x2+2)(f*g)(x) = \sqrt{x + 4} * (5x^2 + 2).
  3. Calculate (f+g)(x)(f+g)(x): Calculate (f+g)(x)=f(x)+g(x)(f+g)(x) = f(x) + g(x).(f+g)(x)=x+4+(5x2+2)(f+g)(x) = \sqrt{x + 4} + (5x^2 + 2).
  4. Determine Domain of f(x)f(x): Determine the domain of f(x)=x+4f(x) = \sqrt{x + 4}.\newlineThe expression under the square root must be non-negative: x+4≥0x + 4 \geq 0.\newlineSo, x≥−4x \geq -4.
  5. Determine Domain of g(x)g(x): Determine the domain of g(x)=5x2+2g(x) = 5x^2 + 2.\newlineSince it's a polynomial, its domain is all real numbers, (−∞,∞)(-\infty, \infty).
  6. Find Domain of (f∗g)(x)(f*g)(x) and (f+g)(x)(f+g)(x): Find the domain of (f∗g)(x)(f*g)(x) and (f+g)(x)(f+g)(x). Both are restricted by the domain of f(x)f(x), since g(x)g(x) has no restrictions. Domain of both (f∗g)(x)(f*g)(x) and (f+g)(x)(f+g)(x) is [−4,∞)[−4, \infty).

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