Let's solve each problem one by one from the image you've uploaded:

Problem 4:

We are given $f(x) = 3x^2 - 2x + 5$. The goal is to find $f(a+b)$.

1. Substitute $a+b$ into the function for $x$: $f(a+b) = 3(a+b)^2 - 2(a+b) + 5$
2. Expand the expression: $(a+b)^2 = a^2 + 2ab + b^2$ So, we get: $f(a+b) = 3(a^2 + 2ab + b^2) - 2(a + b) + 5$
3. Distribute the constants: $f(a+b) = 3a^2 + 6ab + 3b^2 - 2a - 2b + 5$ Thus, the final expression for $f(a+b)$ is: $f(a+b) = 3a^2 + 6ab + 3b^2 - 2a - 2b + 5$

Problem 5:

We are given $f(x) = x^3 - 3x^2 + x - 5$ and asked to find $f\left( -\frac{3}{5} \right)$.

1. Substitute $x = -\frac{3}{5}$ into the function: $f\left( -\frac{3}{5} \right) = \left( -\frac{3}{5} \right)^3 - 3\left( -\frac{3}{5} \right)^2 + \left( -\frac{3}{5} \right) - 5$
2. Calculate each term: $\left( -\frac{3}{5} \right)^3 = -\frac{27}{125}$ $\left( -\frac{3}{5} \right)^2 = \frac{9}{25}, \quad 3\left( \frac{9}{25} \right) = \frac{27}{25}$ $\left( -\frac{3}{5} \right) = -\frac{3}{5}$
3. Combine the terms: $f\left( -\frac{3}{5} \right) = -\frac{27}{125} - \frac{27}{25} - \frac{3}{5} - 5$
4. Convert all terms to a common denominator (125): $f\left( -\frac{3}{5} \right) = -\frac{27}{125} - \frac{135}{125} - \frac{75}{125} - \frac{625}{125}$
5. Combine the fractions: $f\left( -\frac{3}{5} \right) = \frac{-27 - 135 - 75 - 625}{125} = \frac{-862}{125}$ Thus, the result is: $f\left( -\frac{3}{5} \right) = -\frac{862}{125}$

Problem 6:

We are given $f(x) = 4x^5 - x^3 + 5x^2 + \frac{3x}{2} - \frac{1}{2}$ and asked to find $f(3)$.

1. Substitute $x = 3$ into the function: $f(3) = 4(3)^5 - (3)^3 + 5(3)^2 + \frac{3(3)}{2} - \frac{1}{2}$
2. Calculate each term: $3^5 = 243, \quad 4(243) = 972$ $3^3 = 27$ $3^2 = 9, \quad 5(9) = 45$ $\frac{3(3)}{2} = \frac{9}{2}$ $\frac{1}{2} = \frac{1}{2}$
3. Combine the terms: $f(3) = 972 - 27 + 45 + \frac{9}{2} - \frac{1}{2}$ $f(3) = 990 - 27 + \frac{8}{2}$ $f(3) = 963 + 4 = 967$ Thus, the result is: $f(3) = 967$

Would you like more details on any of the solutions? Here are 5 related questions:

1. How can we apply function transformations to evaluate values like $f(a+b)$ for more complex functions?
2. What steps can be taken to simplify polynomial functions further?
3. How do we handle fractional exponents in polynomial evaluation?
4. What are the general methods for solving polynomials with multiple variables?
5. How does substitution work in higher-degree polynomials?

Tip: Always check your calculations step-by-step, especially when dealing with fractional powers or combining terms from different denominators.