I see the image contains several mathematical problems. Let me break down and solve each one.

1. $g(x) = 5x - 7$

• We are asked to evaluate $g(x+1)$.
• Substitute $x+1$ into the function $g(x)$: $g(x+1) = 5(x+1) - 7 = 5x + 5 - 7 = 5x - 2$
• Answer: $g(x+1) = 5x - 2$.

2. $h(x) = \sqrt{x^2 + 2x + 4}$

• Evaluate $h(2)$: $h(2) = \sqrt{2^2 + 2(2) + 4} = \sqrt{4 + 4 + 4} = \sqrt{12} = 2\sqrt{3}$
• Answer: $h(2) = 2\sqrt{3}$.

3. $k(x) = \frac{3x - 2}{2x + 4}$

• Evaluate $k(-3)$: $k(-3) = \frac{3(-3) - 2}{2(-3) + 4} = \frac{-9 - 2}{-6 + 4} = \frac{-11}{-2} = \frac{11}{2}$
• Answer: $k(-3) = \frac{11}{2}$.

4. $f(x) = 2x^2 + 5x - 9$

• Evaluate $f(5x - 2)$: $f(5x - 2) = 2(5x - 2)^2 + 5(5x - 2) - 9$ First, expand $(5x - 2)^2$: $(5x - 2)^2 = 25x^2 - 20x + 4$ Now substitute and simplify: $f(5x - 2) = 2(25x^2 - 20x + 4) + 5(5x - 2) - 9$ $f(5x - 2) = 50x^2 - 40x + 8 + 25x - 10 - 9$ $f(5x - 2) = 50x^2 - 15x - 11$
• Answer: $f(5x - 2) = 50x^2 - 15x - 11$.

5. $g(p) = 4^p$

• Evaluate $g\left( p \cdot \frac{3}{2} \right)$. $g\left( p \cdot \frac{3}{2} \right) = 4^{p \cdot \frac{3}{2}} = (4^p)^{\frac{3}{2}} = 4^{3p/2}$
• Answer: $g\left( p \cdot \frac{3}{2} \right) = 4^{3p/2}$.

Would you like further details on any specific problem?

Here are some related questions for practice:

1. Can you generalize the evaluation for any function $g(x+c)$ given $g(x) = ax + b$?
2. What is the domain and range of $h(x) = \sqrt{x^2 + 2x + 4}$?
3. For the function $k(x)$, what values of $x$ would make the function undefined?
4. Expand and simplify $f(3x - 1)$ if $f(x) = 2x^2 + 5x - 9$.
5. What happens to $g(p)$ when $p = 0$?

Tip:

When simplifying functions, always expand carefully and check for factorizations that might make calculations simpler.