Let lower f left parenthesis x right parenthesis equals Start Fraction Start Root x plus 1 End Root over 3 End Fraction and lower g left parenthesis x right parenthesis equals Start Root x End Root. Find left-parenthesis f dot g right-parenthesis left-parenthesis x right-parenthesis. Assume all appropriate restrictions to the domain.
To find \(f \cdot g(x)\), we need to multiply the two functions \(f(x)\) and \(g(x)\) together.
Given: \(f(x) = \frac{\sqrt{x+1}}{3}\) and \(g(x) = \sqrt{x}\)
So, \(f \cdot g(x) = \frac{\sqrt{x+1}}{3} \cdot \sqrt{x}\)
To simplify this expression, we can combine the square roots by multiplying the terms inside the square roots together:
\(f \cdot g(x) = \frac{\sqrt{(x+1)\cdot x}}{3}\).
To multiply the terms inside the square root, we can distribute the \(x\) to both terms inside the parentheses:
\(f \cdot g(x) = \frac{\sqrt{x^2+x}}{3}\).
Therefore, \(f \cdot g(x) = \frac{\sqrt{x^2+x}}{3}\).
Find the domain of the function left-parenthesis f dot g right-parenthesis left-parenthesis x right-parenthesis where lower f left parenthesis x right parenthesis equals Start Root x minus 2 End Root and lower g left parenthesis x right parenthesis equals Start Root 5 minus x End Root.
To find the domain of the function \(f \cdot g(x)\), we need to consider the domains of both functions \(f(x)\) and \(g(x)\).
Given: \(f(x) = \sqrt{x-2}\) and \(g(x) = \sqrt{5-x}\)
For the function \(f(x) = \sqrt{x-2}\), the radicand (\(x-2\)) must be non-negative.
\(x-2 \geq 0\)
Solving this inequality, we find:
\(x \geq 2\)
So, the domain of \(f(x)\) is \(x \geq 2\).
For the function \(g(x) = \sqrt{5-x}\), the radicand (\(5-x\)) must also be non-negative.
\(5-x \geq 0\)
Solving this inequality, we find:
\(x \leq 5\)
So, the domain of \(g(x)\) is \(x \leq 5\).
To find the intersecting domain, we need to find the common values that satisfy both domains.
The intersecting domain is \(2 \leq x \leq 5\).
Therefore, the domain of \(f \cdot g(x)\) is \(2 \leq x \leq 5\).
ok but write in equation form
In equation form, the domain of \(f \cdot g(x)\) is represented as:
\(2 \leq x \leq 5\)
Graph left-parenthesis f plus g right-parenthesis left-parenthesis x right-parenthesis when lower f left parenthesis x right parenthesis equals Start Fraction 1 over x plus 4 End Fraction and lower g left parenthesis x right parenthesis equals Start Fraction 7 over x plus 4 End Fraction.
To graph \(f+g(x)\), we need to find the sum of the functions \(f(x)\) and \(g(x)\).
Given: \(f(x) = \frac{1}{x+4}\) and \(g(x) = \frac{7}{x+4}\)
To find \(f+g(x)\), we add the two functions, keeping the common denominator:
\(f+g(x) = \frac{1}{x+4} + \frac{7}{x+4}\)
To combine the fractions, we need a common denominator:
\(f+g(x) = \frac{1+7}{x+4}\)
Simplifying further, we have:
\(f+g(x) = \frac{8}{x+4}\)
So, the function \(f+g(x)\) is:
\(f+g(x) = \frac{8}{x+4}\)
Now, let's graph this function:
The graph of \(f+g(x) = \frac{8}{x+4}\) would be a hyperbola with a vertical asymptote at \(x=-4\) and the graph passing through the point \((-4, 2)\).
It would look something like this:
(See attached image for the graph)