Step 1 :The sum of the functions \(f\) and \(g\) is given by \((f+g)(x) = f(x) + g(x)\). Substituting the given functions into the equation, we get \((f+g)(x) = \sqrt{4x - 5} + (2x - 1)\).
Step 2 :The product of the functions \(f\) and \(g\) is given by \((f \cdot g)(x) = f(x) \cdot g(x)\). Substituting the given functions into the equation, we get \((f \cdot g)(x) = \sqrt{4x - 5} \cdot (2x - 1)\).
Step 3 :For the function \(f+g\), the domain is the set of all x-values that make the expression under the square root in \(f(x)\) non-negative, and also make the expression in \(g(x)\) defined. Since \(g(x)\) is a linear function, it is defined for all real numbers. However, for \(f(x)\), we must have \(4x - 5 \geq 0\). Solving this inequality gives \(x \geq \frac{5}{4}\). So, the domain of \(f+g\) is \([\frac{5}{4}, \infty)\).
Step 4 :For the function \(f \cdot g\), the domain is also the set of all x-values that make the expression under the square root in \(f(x)\) non-negative, and also make the expression in \(g(x)\) defined. Again, since \(g(x)\) is a linear function, it is defined for all real numbers. And for \(f(x)\), we must have \(4x - 5 \geq 0\). Solving this inequality gives \(x \geq \frac{5}{4}\). So, the domain of \(f \cdot g\) is also \([\frac{5}{4}, \infty)\).