1. **Stating the problem:** We analyze the function $f_1: y = x - 3$. 2. **Domain and range:** The function $y = x - 3$ is a linear function defined for all real numbers, so the domain is $\mathbb{R}$. The range is also $\mathbb{R}$ because as $x$ takes any real value, $y$ can take any real value. 3. **Intercepts:** - To find the $x$-intercept, set $y=0$: $$0 = x - 3 \implies x = 3$$ So the $x$-intercept is at $(3,0)$. - To find the $y$-intercept, set $x=0$: $$y = 0 - 3 = -3$$ So the $y$-intercept is at $(0,-3)$. 4. **Monotonicity:** The function is linear with slope $1 > 0$, so it is strictly increasing (monotonically increasing) on $\mathbb{R}$. 5. **Extrema:** Since the function is strictly increasing and linear, it has no local maxima or minima. 6. **Boundedness:** The function is unbounded above and below. 7. **Parity:** Check if $f(-x) = f(x)$ or $f(-x) = -f(x)$: $$f(-x) = -x - 3 \neq f(x) = x - 3$$ $$f(-x) = -x - 3 \neq -(x - 3) = -x + 3$$ So the function is neither even nor odd. 8. **Injectivity and invertibility:** The function is strictly increasing and thus injective, so it has an inverse. 9. **Finding the inverse function:** Start with $$y = x - 3$$ Swap $x$ and $y$: $$x = y - 3$$ Solve for $y$: $$y = x + 3$$ So the inverse function is: $$f_1^{-1}(x) = x + 3$$ 10. **Domain and range of inverse:** Since the original function has domain $\mathbb{R}$ and range $\mathbb{R}$, the inverse also has domain $\mathbb{R}$ and range $\mathbb{R}$. 11. **Intercepts of inverse:** - $x$-intercept: set $y=0$: $$0 = x + 3 \implies x = -3$$ - $y$-intercept: set $x=0$: $$y = 0 + 3 = 3$$ So the inverse intercepts are $(-3,0)$ and $(0,3)$. 12. **Intersection of $f$ and $f^{-1}$:** Solve $f(x) = f^{-1}(x)$: $$x - 3 = x + 3$$ This is never true, so no intersection points. However, the function and its inverse always intersect on the line $y=x$. Find points where $f(x) = x$: $$x - 3 = x \implies -3 = 0$$ No solution. Find points where $f^{-1}(x) = x$: $$x + 3 = x \implies 3 = 0$$ No solution. But the function and its inverse are reflections about $y=x$. **Final answer:** - Domain: $\mathbb{R}$ - Range: $\mathbb{R}$ - $x$-intercept: $(3,0)$ - $y$-intercept: $(0,-3)$ - Monotonically increasing - No extrema - Unbounded - Neither even nor odd - Inverse function: $f_1^{-1}(x) = x + 3$ - Inverse domain and range: $\mathbb{R}$ - Inverse intercepts: $(-3,0)$ and $(0,3)$ - No intersection points of $f$ and $f^{-1}$ except reflection symmetry about $y=x$.