Use the functions f left parenthesis x right parenthesis equals x squared minus 4 and g left parenthesis x right parenthesis equals x plus 8 to answer parts​ (a)-(g).
​(a) Solve f left parenthesis x right parenthesis equals 0.
​(d) Solve f left parenthesis x right parenthesis greater than 0.
​(g) Solve f left parenthesis x right parenthesis greater than or equals 1.
​(b) Solve g left parenthesis x right parenthesis equals 0.
​(e) Solve g left parenthesis x right parenthesis less than or equals 0.
​(c) Solve f left parenthesis x right parenthesis equals g left parenthesis x right parenthesis.
​(f) Solve f left parenthesis x right parenthesis greater than g left parenthesis x right parenthesis.

To answer these parts using the given functions \( f(x) = x^2 - 4 \) and \( g(x) = x + 8 \), let’s break down each one step by step.

### (a) Solve \( f(x) = 0 \)
We are asked to solve \( x^2 - 4 = 0 \).

1. Factor the quadratic equation:
   \[
   (x - 2)(x + 2) = 0
   \]
2. Set each factor equal to zero:
   \[
   x - 2 = 0 \quad \text{or} \quad x + 2 = 0
   \]
   \[
   x = 2 \quad \text{or} \quad x = -2
   \]
Thus, the solutions are \( x = 2 \) and \( x = -2 \).

### (d) Solve \( f(x) > 0 \)
We are asked to solve \( x^2 - 4 > 0 \).

1. Factor the inequality:
   \[
   (x - 2)(x + 2) > 0
   \]
2. The critical points are \( x = 2 \) and \( x = -2 \). We test intervals:
   - For \( x < -2 \), choose \( x = -3 \): \( (-3)^2 - 4 = 9 - 4 = 5 \), which is positive.
   - For \( -2 < x < 2 \), choose \( x = 0 \): \( (0)^2 - 4 = -4 \), which is negative.
   - For \( x > 2 \), choose \( x = 3 \): \( (3)^2 - 4 = 9 - 4 = 5 \), which is positive.

Therefore, \( f(x) > 0 \) when \( x < -2 \) or \( x > 2 \). In interval notation, the solution is:
\[
x \in (-\infty, -2) \cup (2, \infty)
\]

### (g) Solve \( f(x) \geq 1 \)
We are asked to solve \( x^2 - 4 \geq 1 \).

1. Add 4 to both sides:
   \[
   x^2 \geq 5
   \]
2. Solve for \( x \):
   \[
   x \geq \sqrt{5} \quad \text{or} \quad x \leq -\sqrt{5}
   \]
Thus, the solution is \( x \in (-\infty, -\sqrt{5}] \cup [\sqrt{5}, \infty) \).

### (b) Solve \( g(x) = 0 \)
We are asked to solve \( x + 8 = 0 \).

1. Subtract 8 from both sides:
   \[
   x = -8
   \]
Thus, the solution is \( x = -8 \).

### (e) Solve \( g(x) \leq 0 \)
We are asked to solve \( x + 8 \leq 0 \).

1. Subtract 8 from both sides:
   \[
   x \leq -8
   \]
Thus, the solution is \( x \leq -8 \).

### (c) Solve \( f(x) = g(x) \)
We are asked to solve \( x^2 - 4 = x + 8 \).

1. Rearrange the equation:
   \[
   x^2 - x - 12 = 0
   \]
2. Factor the quadratic equation:
   \[
   (x - 4)(x + 3) = 0
   \]
3. Set each factor equal to zero:
   \[
   x = 4 \quad \text{or} \quad x = -3
   \]
Thus, the solutions are \( x = 4 \) and \( x = -3 \).

### (f) Solve \( f(x) > g(x) \)
We are asked to solve \( x^2 - 4 > x + 8 \).

1. Rearrange the inequality:
   \[
   x^2 - x - 12 > 0
   \]
2. Factor the quadratic inequality:
   \[
   (x - 4)(x + 3) > 0
   \]
3. Test intervals:
   - For \( x < -3 \), choose \( x = -4 \): \( (-4 - 4)(-4 + 3) = (-8)(-1) > 0 \).
   - For \( -3 < x < 4 \), choose \( x = 0 \): \( (0 - 4)(0 + 3) = (-4)(3) < 0 \).
   - For \( x > 4 \), choose \( x = 5 \): \( (5 - 4)(5 + 3) = (1)(8) > 0 \).

Thus, \( f(x) > g(x) \) when \( x < -3 \) or \( x > 4 \). In interval notation, the solution is:
\[
x \in (-\infty, -3) \cup (4, \infty)
\]

---

Would you like more details on any part, or do you have any questions? Here are some related problems to explore:

1. Solve \( f(x) = 1 \) for \( f(x) = x^2 - 4 \).
2. Find the range of \( g(x) = x + 8 \).
3. Solve \( f(x) \leq 0 \).
4. Determine when \( f(x) = g(x) + 5 \).
5. Analyze the graph of \( f(x) \) and \( g(x) \) to visualize the solutions.

**Tip**: When solving inequalities involving quadratics, always test intervals based on critical points found by solving the equation first.

Use the functions f(x) = x^2 - 4 and g(x) = x + 8 to answer parts (a)-(g): (a) Solve f(x) = 0. (d) Solve f(x) > 0. (g) Solve f(x) ≥ 1. (b) Solve g(x) = 0. (e) Solve g(x) ≤ 0. (c) Solve f(x) = g(x). (f) Solve f(x) > g(x).

Learn how to solve quadratic and linear equations using f(x) = x^2 - 4 and g(x) = x + 8. This detailed step-by-step guide covers solving equations and inequalities involving quadratic and linear functions, ideal for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation