Answer :
To determine the correct order of the functions based on their minimum values, we'll evaluate each function separately and compare their minimum values.
### Step 1: Evaluate [tex]\( f(x) \)[/tex]
The given function is:
[tex]\[ f(x) = (x + 2)^2 - 7 \][/tex]
To find the minimum value of [tex]\( f(x) \)[/tex]:
1. Recognize that [tex]\((x + 2)^2\)[/tex] is a quadratic function that achieves its minimum value of 0 when [tex]\( x = -2 \)[/tex].
2. Plug [tex]\( x = -2 \)[/tex] into the function:
[tex]\[ f(-2) = ((-2) + 2)^2 - 7 = 0 - 7 = -7 \][/tex]
So, the minimum value of [tex]\( f(x) \)[/tex] is [tex]\(-7\)[/tex].
### Step 2: Evaluate [tex]\( g(x) \)[/tex]
We are given specific values of [tex]\( g(x) \)[/tex] at certain points:
[tex]\[ \begin{array}{|c|c|} \hline x & g(x) \\ \hline 0 & 4 \\ \hline 1 & -1 \\ \hline 2 & -4 \\ \hline 3 & -5 \\ \hline 4 & -4 \\ \hline 5 & -1 \\ \hline 6 & 4 \\ \hline \end{array} \][/tex]
The smallest value among these values of [tex]\( g(x) \)[/tex] is [tex]\(-5\)[/tex].
### Step 3: Evaluate [tex]\( h(x) \)[/tex]
Assume [tex]\( h(x) = x^2 \)[/tex].
To find the minimum value of [tex]\( h(x) \)[/tex]:
1. Recognize that [tex]\( x^2 \)[/tex] is a quadratic function that achieves its minimum value of 0 when [tex]\( x = 0 \)[/tex].
2. Plug [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ h(0) = 0^2 = 0 \][/tex]
So, the minimum value of [tex]\( h(x) \)[/tex] is [tex]\( 0 \)[/tex].
### Step 4: Compare and Order the Minimum Values
Now we have:
- Minimum value of [tex]\( f(x) \)[/tex] = -7
- Minimum value of [tex]\( g(x) \)[/tex] = -5
- Minimum value of [tex]\( h(x) \)[/tex] = 0
Comparing these values from smallest to largest gives us the order:
[tex]\[ f(x), g(x), h(x) \][/tex]
### Answer
The correct order of the functions from smallest minimum value to largest minimum value is:
[tex]\[ f(x), g(x), h(x) \][/tex]
### Step 1: Evaluate [tex]\( f(x) \)[/tex]
The given function is:
[tex]\[ f(x) = (x + 2)^2 - 7 \][/tex]
To find the minimum value of [tex]\( f(x) \)[/tex]:
1. Recognize that [tex]\((x + 2)^2\)[/tex] is a quadratic function that achieves its minimum value of 0 when [tex]\( x = -2 \)[/tex].
2. Plug [tex]\( x = -2 \)[/tex] into the function:
[tex]\[ f(-2) = ((-2) + 2)^2 - 7 = 0 - 7 = -7 \][/tex]
So, the minimum value of [tex]\( f(x) \)[/tex] is [tex]\(-7\)[/tex].
### Step 2: Evaluate [tex]\( g(x) \)[/tex]
We are given specific values of [tex]\( g(x) \)[/tex] at certain points:
[tex]\[ \begin{array}{|c|c|} \hline x & g(x) \\ \hline 0 & 4 \\ \hline 1 & -1 \\ \hline 2 & -4 \\ \hline 3 & -5 \\ \hline 4 & -4 \\ \hline 5 & -1 \\ \hline 6 & 4 \\ \hline \end{array} \][/tex]
The smallest value among these values of [tex]\( g(x) \)[/tex] is [tex]\(-5\)[/tex].
### Step 3: Evaluate [tex]\( h(x) \)[/tex]
Assume [tex]\( h(x) = x^2 \)[/tex].
To find the minimum value of [tex]\( h(x) \)[/tex]:
1. Recognize that [tex]\( x^2 \)[/tex] is a quadratic function that achieves its minimum value of 0 when [tex]\( x = 0 \)[/tex].
2. Plug [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ h(0) = 0^2 = 0 \][/tex]
So, the minimum value of [tex]\( h(x) \)[/tex] is [tex]\( 0 \)[/tex].
### Step 4: Compare and Order the Minimum Values
Now we have:
- Minimum value of [tex]\( f(x) \)[/tex] = -7
- Minimum value of [tex]\( g(x) \)[/tex] = -5
- Minimum value of [tex]\( h(x) \)[/tex] = 0
Comparing these values from smallest to largest gives us the order:
[tex]\[ f(x), g(x), h(x) \][/tex]
### Answer
The correct order of the functions from smallest minimum value to largest minimum value is:
[tex]\[ f(x), g(x), h(x) \][/tex]
