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The problem provides two functions, \( f(x) = -x^2 + 4 \) and \( g(x) = 4x + 4 \), and asks us to solve a series of inequalities and equations involving these functions. Let's go through each part step-by-step:

### (a) Solve \( f(x) = 0 \)

To solve \( f(x) = 0 \):

\[
f(x) = -x^2 + 4 = 0
\]

\[
-x^2 + 4 = 0 \implies x^2 = 4
\]

\[
x = \pm 2
\]

### (b) Solve \( g(x) = 0 \)

To solve \( g(x) = 0 \):

\[
g(x) = 4x + 4 = 0
\]

\[
4x + 4 = 0 \implies 4x = -4
\]

\[
x = -1
\]

### (c) Solve \( f(x) = g(x) \)

To solve \( f(x) = g(x) \):

\[
-x^2 + 4 = 4x + 4
\]

Simplify:

\[
-x^2 - 4x = 0
\]

Factor out:

\[
-x(x + 4) = 0
\]

\[
x = 0 \text{ or } x = -4
\]

### (d) Solve \( f(x) > 0 \)

Solve \( -x^2 + 4 > 0 \):

\[
x^2 < 4 \implies -2 < x < 2
\]

### (e) Solve \( g(x) \leq 0 \)

Solve \( 4x + 4 \leq 0 \):

\[
4x \leq -4 \implies x \leq -1
\]

### (f) Solve \( f(x) > g(x) \)

Solve \( -x^2 + 4 > 4x + 4 \):

\[
-x^2 - 4x < 0
\]

\[
x(x + 4) > 0
\]

Interval test gives:

\[
x < -4 \text{ or } x > 0
\]

### (g) Solve \( f(x) \geq 4 \)

Solve \( -x^2 + 4 \geq 4 \):

\[
x^2 \leq 0
\]

\( x = 0 \)

Would you like further details or have any questions?

### Related Questions:
1. How do you solve inequalities with quadratic functions?
2. What are the solutions to a function intersecting another line?
3. Can \( x^2 \) have negative values? If not, why?
4. How do you determine where one function is greater than another?
5. What steps should you take when factoring polynomials?

**Tip:** When solving inequalities, sketching graphs can help visualize the solutions.

Use the functions f(x) = -x^2 + 4 and g(x) = 4x + 4 to answer parts (a)-(g): (a) Solve f(x) = 0. (b) Solve g(x) = 0. (c) Solve f(x) = g(x). (d) Solve f(x) > 0. (e) Solve g(x) <= 0. (f) Solve f(x) > g(x). (g) Solve f(x) >= 4.

Learn how to solve various types of equations and inequalities using the functions f(x) = -x^2 + 4 and g(x) = 4x + 4. This problem includes finding solutions to quadratic and linear equations and analyzing inequalities involving these functions. Suitable for students in Grades 9-11.

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