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For problem 27, we are asked to solve several inequalities and equations using the given functions \( f(x) = -x^2 + 1 \) and \( g(x) = 4x + 1 \).

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

We start by solving for \( f(x) = 0 \):
\[
f(x) = -x^2 + 1 = 0
\]
\[
-x^2 + 1 = 0 \implies x^2 = 1
\]
\[
x = \pm 1
\]
So the solutions are \( x = 1 \) and \( x = -1 \).

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

Next, we solve for \( g(x) = 0 \):
\[
g(x) = 4x + 1 = 0
\]
\[
4x = -1 \implies x = -\frac{1}{4}
\]
Thus, \( x = -\frac{1}{4} \).

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

We now solve \( f(x) = g(x) \):
\[
-x^2 + 1 = 4x + 1
\]
Subtract 1 from both sides:
\[
-x^2 = 4x
\]
Move all terms to one side:
\[
x^2 + 4x = 0
\]
Factor the equation:
\[
x(x + 4) = 0
\]
So, \( x = 0 \) or \( x = -4 \).

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

We now solve \( f(x) > 0 \):
\[
-x^2 + 1 > 0
\]
\[
-x^2 > -1 \implies x^2 < 1
\]
So \( -1 < x < 1 \). The solution is \( x \in (-1, 1) \).

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

Next, solve \( g(x) \leq 0 \):
\[
4x + 1 \leq 0
\]
\[
4x \leq -1 \implies x \leq -\frac{1}{4}
\]
Thus, \( x \leq -\frac{1}{4} \).

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

Finally, we solve \( f(x) > g(x) \):
\[
-x^2 + 1 > 4x + 1
\]
Subtract 1 from both sides:
\[
-x^2 > 4x
\]
Move all terms to one side:
\[
x^2 + 4x < 0
\]
Factor the inequality:
\[
x(x + 4) < 0
\]
Using a sign chart or testing intervals, we find that the solution is \( -4 < x < 0 \).

---

Would you like more details on any step or a deeper explanation on a particular part?

Here are 5 related questions to expand your understanding:

1. How do we solve quadratic inequalities like \( f(x) > g(x) \)?
2. What are the steps for solving quadratic equations like \( f(x) = 0 \)?
3. Why do we factor expressions like \( x^2 + 4x < 0 \)?
4. How do sign charts help in solving inequalities?
5. How can we interpret the graphical intersection of \( f(x) \) and \( g(x) \)?

**Tip:** When solving inequalities, always check the boundary points by substituting them into the inequality.

Solve the following equations and inequalities using f(x) = -x^2 + 1 and g(x) = 4x + 1

Learn how to solve equations and inequalities involving quadratic functions f(x) = -x^2 + 1 and g(x) = 4x + 1. This step-by-step guide explores solutions for equalities, inequalities, and intersections of functions. Suitable for students in Grades 9-12.

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