To find the compositions of the functions given the definitions:
- \( f(x) = x^2 \)
- \( h(x) = x + 9 \)
- \( p(x) = \frac{\sqrt{x}}{x} + 4 \)
- \( v(x) = 5x + 3 \)
Let's evaluate each composition step-by-step:
A. \( V(f(x)) \)
- First, find \( f(x) \): \[ f(x) = x^2 \]
- Next, substitute \( f(x) \) into \( v(x) \): \[ V(f(x)) = V(x^2) = 5(x^2) + 3 = 5x^2 + 3 \]
B. \( h(p(16)) \)
- First, calculate \( p(16) \): \[ p(16) = \frac{\sqrt{16}}{16} + 4 = \frac{4}{16} + 4 = \frac{1}{4} + 4 = \frac{1}{4} + \frac{16}{4} = \frac{17}{4} \]
- Now substitute \( p(16) \) into \( h(x) \): \[ h\left(\frac{17}{4}\right) = \frac{17}{4} + 9 = \frac{17}{4} + \frac{36}{4} = \frac{53}{4} \]
C. \( V(p(x)) \)
- Next, substitute \( p(x) \) into \( v(x) \): \[ V(p(x)) = V\left(\frac{\sqrt{x}}{x} + 4\right) = 5\left(\frac{\sqrt{x}}{x} + 4\right) + 3 \] \[ = 5\cdot \frac{\sqrt{x}}{x} + 20 + 3 = \frac{5\sqrt{x}}{x} + 23 \]
D. \( h(v(1)) \)
- First, calculate \( v(1) \): \[ v(1) = 5(1) + 3 = 5 + 3 = 8 \]
- Then substitute \( v(1) \) into \( h(x) \): \[ h(8) = 8 + 9 = 17 \]
E. \( h(f(x)) \)
- Substitute \( f(x) \) into \( h(x) \): \[ h(f(x)) = h(x^2) = x^2 + 9 \]
F. \( V(h(f(x))) \)
- We already found \( h(f(x)) = x^2 + 9 \).
- Now substitute this into \( v(x) \): \[ V(h(f(x))) = V(x^2 + 9) = 5(x^2 + 9) + 3 = 5x^2 + 45 + 3 = 5x^2 + 48 \]
Finally, summarizing the results:
- A. \( V(f(x)) = 5x^2 + 3 \)
- B. \( h(p(16)) = \frac{53}{4} \)
- C. \( V(p(x)) = \frac{5\sqrt{x}}{x} + 23 \)
- D. \( h(v(1)) = 17 \)
- E. \( h(f(x)) = x^2 + 9 \)
- F. \( V(h(f(x))) = 5x^2 + 48 \)