Question

f(x)=x^2
g(x)=x
Find the values of x when f(x) is greater than g(x).

Answer 1

To find the values of \( x \) where \( f(x) > g(x) \), we need to solve the inequality:

\[
f(x) = x^2\quad \text{and} \quad g(x) = x
\]

We want to find when:

\[
x^2 > x
\]

First, we can rearrange the inequality:

\[
x^2 - x > 0
\]

Next, we factor the left-hand side:

\[
x(x - 1) > 0
\]

Now, we need to determine the intervals where this inequality holds. To do this, we find the roots of the equation \( x(x - 1) = 0 \), which gives us:

\[
x = 0 \quad \text{and} \quad x = 1
\]

These roots divide the number line into three intervals:

1. \( (-\infty, 0) \)
2. \( (0, 1) \)
3. \( (1, \infty) \)

Next, we test a point from each interval to see where the product \( x(x - 1) \) is positive.

1. In the interval \( (-\infty, 0) \), let's test \( x = -1 \):
\[
(-1)(-1 - 1) = (-1)(-2) = 2 > 0
\]
So, \( x(x - 1) > 0 \) in this interval.

2. In the interval \( (0, 1) \), let's test \( x = 0.5 \):
\[
(0.5)(0.5 - 1) = (0.5)(-0.5) = -0.25 < 0
\]
So, \( x(x - 1) < 0 \) in this interval.

3. In the interval \( (1, \infty) \), let's test \( x = 2 \):
\[
(2)(2 - 1) = (2)(1) = 2 > 0
\]
So, \( x(x - 1) > 0 \) in this interval.

Now, we summarize the results:

- \( f(x) > g(x) \) in the intervals \( (-\infty, 0) \) and \( (1, \infty) \).
- \( f(x) = g(x) \) at the points \( x = 0 \) and \( x = 1 \).
- \( f(x) < g(x) \) in the interval \( (0, 1) \).

Therefore, the values of \( x \) for which \( f(x) > g(x) \) are:

\[
x \in (-\infty, 0) \cup (1, \infty)
\]