1. The problem involves understanding how to map or transform the function $f(x)$ to get $g(x)$ using given formulas. 2. For part (c), we have: $$f(x) = \sqrt{x}$$ $$g(x) = \frac{1}{2} f \left( \frac{1}{2} (x + 3) \right) + 5$$ This means we first replace $x$ in $f(x)$ with $\frac{1}{2}(x+3)$, then multiply the result by $\frac{1}{2}$, and finally add 5. 3. Writing $g(x)$ explicitly: $$g(x) = \frac{1}{2} \sqrt{\frac{1}{2} (x + 3)} + 5$$ 4. This transformation involves: - Horizontal scaling by a factor of 2 (because of $\frac{1}{2}$ inside the function argument). - Horizontal shift left by 3 units (because of $x + 3$ inside the function). - Vertical scaling by $\frac{1}{2}$ (outside the square root). - Vertical shift up by 5 units. 5. For part (d), we have: $$f(x) = \frac{1}{x}$$ $$g(x) = 2 f[-(x - 3)] + 4$$ 6. Writing $g(x)$ explicitly: $$g(x) = 2 \left( \frac{1}{-(x - 3)} \right) + 4 = 2 \left( -\frac{1}{x - 3} \right) + 4 = -\frac{2}{x - 3} + 4$$ 7. This transformation involves: - Horizontal shift right by 3 units (because of $x - 3$ inside the function). - Reflection about the y-axis (due to the negative sign inside the function argument). - Vertical scaling by 2. - Vertical shift up by 4 units. 8. To map $f(x)$ to $g(x)$, apply the inside transformations to the input $x$ first (horizontal shifts and scalings), then apply the outside transformations (vertical scalings and shifts). Final answers: - For (c): $$g(x) = \frac{1}{2} \sqrt{\frac{1}{2} (x + 3)} + 5$$ - For (d): $$g(x) = -\frac{2}{x - 3} + 4$$