1. **Problem statement:** We need to find a function $f$ such that it satisfies the properties: $$f(x + 1) = f(x) + 5$$ and $$f(0) = 2$$ 2. **Understanding the property:** The equation $f(x + 1) = f(x) + 5$ means that when we increase the input $x$ by 1, the output increases by 5. This suggests the function is linear with a constant rate of change (slope) of 5. 3. **General form of the function:** A linear function can be written as: $$f(x) = mx + b$$ where $m$ is the slope and $b$ is the y-intercept. 4. **Using the property to find $m$:** From the property, $$f(x + 1) = m(x + 1) + b = mx + m + b$$ and $$f(x) + 5 = mx + b + 5$$ Since these are equal for all $x$, $$mx + m + b = mx + b + 5$$ Subtracting $mx + b$ from both sides, $$m = 5$$ 5. **Using the initial condition to find $b$:** Given $f(0) = 2$, $$f(0) = m imes 0 + b = b = 2$$ 6. **Final function:** $$f(x) = 5x + 2$$ This function satisfies both given properties.