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The question in the image reads:

"29. Given \( f(x) = 2x^2 - 4x \), determine \( f(x) > 0 \) and \( f^{-1}(x) > 0 \). Identify the domain of both \( f(x) \) and \( f^{-1}(x) \)."

Let's break this down step by step.

### 1. Solve \( f(x) > 0 \):

We start with the quadratic function \( f(x) = 2x^2 - 4x \). First, factor it:
\[
f(x) = 2x(x - 2)
\]
To solve \( f(x) > 0 \), we analyze the factored form:
\[
2x(x - 2) > 0
\]
The critical points are \( x = 0 \) and \( x = 2 \). These points divide the number line into three intervals: \( (-\infty, 0) \), \( (0, 2) \), and \( (2, \infty) \).

By testing a point in each interval:
- In \( (-\infty, 0) \), pick \( x = -1 \): \( 2(-1)((-1) - 2) = 2(-1)(-3) = 6 > 0 \).
- In \( (0, 2) \), pick \( x = 1 \): \( 2(1)((1) - 2) = 2(1)(-1) = -2 < 0 \).
- In \( (2, \infty) \), pick \( x = 3 \): \( 2(3)((3) - 2) = 2(3)(1) = 6 > 0 \).

Thus, \( f(x) > 0 \) on the intervals \( (-\infty, 0) \cup (2, \infty) \).

### 2. Solve for \( f^{-1}(x) \):

Next, find the inverse of \( f(x) \). Start by solving \( y = 2x^2 - 4x \) for \( x \). Complete the square:
\[
y = 2(x^2 - 2x) = 2((x - 1)^2 - 1) = 2(x - 1)^2 - 2
\]
Solve for \( x \):
\[
y + 2 = 2(x - 1)^2
\]
\[
\frac{y + 2}{2} = (x - 1)^2
\]
\[
x - 1 = \pm \sqrt{\frac{y + 2}{2}}
\]
Thus,
\[
x = 1 \pm \sqrt{\frac{y + 2}{2}}
\]

Since a quadratic function isn't one-to-one without restricting its domain, \( f^{-1}(x) \) may not be applicable in the entire range unless restrictions are applied.

### 3. Domain of \( f(x) \):

Since \( f(x) = 2x^2 - 4x \) is a quadratic function, its domain is all real numbers, \( (-\infty, \infty) \).

### 4. Domain of \( f^{-1}(x) \):

The inverse function \( f^{-1}(x) \) will only exist for \( x \geq -2 \) because of the square root term in its formula. Therefore, the domain of \( f^{-1}(x) \) is \( [-2, \infty) \).

Would you like any details or further clarifications?

### Related Questions:
1. How can we restrict the domain of \( f(x) \) to ensure it has an inverse?
2. How can we determine the range of a quadratic function like \( f(x) \)?
3. What is the graphical interpretation of \( f(x) > 0 \) and \( f^{-1}(x) > 0 \)?
4. How does completing the square help in finding the inverse of a quadratic function?
5. How can we verify if an inverse function exists for any given function?

**Tip:** When solving quadratic inequalities, always test points in each interval defined by the critical points.

Given f(x) = 2x^2 - 4x, determine f(x) > 0 and f^-1(x) > 0. Identify the domain of both f(x) and f^-1(x).

Learn how to solve the quadratic inequality f(x) = 2x^2 - 4x > 0 and identify the domain of both the function and its inverse. This problem covers key algebraic concepts such as factorization and inverse functions. Suitable for Grades 11-12.

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