1. **Problem:** For the function $f(x) = 4x - 5$, find: i) The greatest possible domain ii) The range iii) Whether the function is one-to-one or many-to-one 2. **Formula and rules:** - The domain of a function is the set of all input values $x$ for which the function is defined. - The range is the set of all possible output values $f(x)$. - A function is one-to-one if each output corresponds to exactly one input. 3. **Step i) Domain:** - Since $f(x) = 4x - 5$ is a linear function, it is defined for all real numbers. - So, the domain is $\mathbb{R}$. 4. **Step ii) Range:** - A linear function with non-zero slope can produce all real numbers as output. - Therefore, the range is $\mathbb{R}$. 5. **Step iii) One-to-one or many-to-one:** - Since the slope 4 is non-zero, $f$ is strictly increasing. - Hence, $f$ is one-to-one. **Final answers:** - Domain: $\mathbb{R}$ - Range: $\mathbb{R}$ - Function type: One-to-one