Sure! Let's analyze each statement regarding the functions [tex]\( g(x) = x^2 \)[/tex] and [tex]\( f(x) = x \)[/tex]:
1. Statement 1: [tex]\( f(x) \)[/tex] is greater than [tex]\( g(x) \)[/tex] on the interval [tex]\( (0,1) \)[/tex]
To check whether [tex]\( f(x) = x \)[/tex] is greater than [tex]\( g(x) = x^2 \)[/tex] in the interval [tex]\( (0, 1) \)[/tex], we need to compare the values of the functions in this range. For any [tex]\( x \)[/tex] in [tex]\( (0, 1) \)[/tex]:
[tex]\[
x > x^2
\][/tex]
If we analyze this statement thoroughly, we can see that for [tex]\( 0 < x < 1 \)[/tex]:
[tex]\[
x^2 < x
\][/tex]
Therefore, [tex]\( f(x) \)[/tex] is indeed greater than [tex]\( g(x) \)[/tex] on the interval [tex]\( (0, 1) \)[/tex].
Thus, the statement is True.
2. Statement 2: [tex]\( f(-1) \)[/tex] is equal to [tex]\( g(-1) \)[/tex]
Let's compute the values at [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = -1 \quad \text{and} \quad g(-1) = (-1)^2 = 1
\][/tex]
Since
[tex]\[
-1 \neq 1
\][/tex]
Thus, this statement is False.
3. Statement 3: [tex]\( g(x) \)[/tex] has a greater [tex]\( y \)[/tex]-intercept than [tex]\( f(x) \)[/tex] does
To analyze the y-intercepts, we evaluate both functions at [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = 0^2 = 0 \quad \text{and} \quad f(0) = 0
\][/tex]
So,
[tex]\[
g(x) \quad \text{and} \quad f(x) \quad \text{both have a y-intercept of 0}
\][/tex]
Therefore, this statement is False.
4. Statement 4: [tex]\( f(1) \)[/tex] is equal to [tex]\( g(1) \)[/tex]
Let's compute the values at [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 1 \quad \text{and} \quad g(1) = 1^2 = 1
\][/tex]
Since
[tex]\[
1 = 1
\][/tex]
Thus, this statement is True.
So, the correct statements are:
- [tex]\( f(x) \)[/tex] is greater than [tex]\( g(x) \)[/tex] on the interval [tex]\( (0,1) \)[/tex].
- [tex]\( f(1) \)[/tex] is equal to [tex]\( g(1) \)[/tex].
Thus, the results are:
1. True
2. False
3. False
4. True
