Suppose that the functions \( f \) and \( g \) are defined as follows. \[ \begin{array}{l} f(x)=3 x+5 \\ g(x)=\sqrt{5 x-3} \end{array} \] Find \( f \cdot g \) and \( f-g \). Then, give their domains using interval not \[ (f \cdot g)(x)= \] Domain of \( f \cdot g \) : \( \square \) \[ (f-g)(x)= \] \( \square \) Domain of \( f-g \) : \( \square \)
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To find \( f \cdot g \), we multiply the functions \( f \) and \( g \): \[ (f \cdot g)(x) = f(x) \cdot g(x) = (3x + 5) \cdot \sqrt{5x - 3}. \] Now, to determine the domain of \( f \cdot g \), we have to consider where both \( f(x) \) and \( g(x) \) are defined. The function \( g(x) = \sqrt{5x - 3} \) is defined when its argument is non-negative: \[ 5x - 3 \geq 0 \implies x \geq \frac{3}{5}. \] Thus, the domain of \( f \cdot g \) is \( [\frac{3}{5}, \infty) \). Next, we compute \( f - g \): \[ (f - g)(x) = f(x) - g(x) = (3x + 5) - \sqrt{5x - 3}. \] For the domain of \( f - g \), we again need to ensure \( g(x) \) is defined, so we still require: \[ 5x - 3 \geq 0 \implies x \geq \frac{3}{5}. \] Therefore, the domain of \( f - g \) is also \( [\frac{3}{5}, \infty) \). Now we can summarize the results: \[ (f \cdot g)(x) = (3x + 5) \cdot \sqrt{5x - 3} \] Domain of \( f \cdot g \): \( [\frac{3}{5}, \infty) \) \[ (f - g)(x) = (3x + 5) - \sqrt{5x - 3} \] Domain of \( f - g \): \( [\frac{3}{5}, \infty) \)
