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Answer :
To determine the values of [tex]\( f(-2) \)[/tex], [tex]\( f(0) \)[/tex], and [tex]\( f(4) \)[/tex] from the piecewise function, we need to evaluate each case based on the conditions given in the function.
The piecewise function is defined as:
1. [tex]\( f(x) = 2x + 5 \)[/tex] when [tex]\( x < -1 \)[/tex]
2. [tex]\( f(x) = 4x^2 + 2 \)[/tex] when [tex]\(-1 < x \leq 3 \)[/tex]
3. [tex]\( f(x) = 6x^2 - 2 \)[/tex] when [tex]\( x > 3 \)[/tex]
Now let's evaluate each specified value:
### Finding [tex]\( f(-2) \)[/tex]
- Since [tex]\(-2 < -1\)[/tex], we use the formula [tex]\( f(x) = 2x + 5 \)[/tex].
[tex]\[
f(-2) = 2(-2) + 5 = -4 + 5 = 1
\][/tex]
### Finding [tex]\( f(0) \)[/tex]
- The value 0 fits in the condition [tex]\(-1 < 0 \leq 3\)[/tex], so we use the formula [tex]\( f(x) = 4x^2 + 2 \)[/tex].
[tex]\[
f(0) = 4(0)^2 + 2 = 0 + 2 = 2
\][/tex]
### Finding [tex]\( f(4) \)[/tex]
- Since [tex]\( 4 > 3 \)[/tex], we use the formula [tex]\( f(x) = 6x^2 - 2 \)[/tex].
[tex]\[
f(4) = 6(4)^2 - 2 = 6(16) - 2 = 96 - 2 = 94
\][/tex]
In conclusion, the values are:
- [tex]\( f(-2) = 1 \)[/tex]
- [tex]\( f(0) = 2 \)[/tex]
- [tex]\( f(4) = 94 \)[/tex]
The piecewise function is defined as:
1. [tex]\( f(x) = 2x + 5 \)[/tex] when [tex]\( x < -1 \)[/tex]
2. [tex]\( f(x) = 4x^2 + 2 \)[/tex] when [tex]\(-1 < x \leq 3 \)[/tex]
3. [tex]\( f(x) = 6x^2 - 2 \)[/tex] when [tex]\( x > 3 \)[/tex]
Now let's evaluate each specified value:
### Finding [tex]\( f(-2) \)[/tex]
- Since [tex]\(-2 < -1\)[/tex], we use the formula [tex]\( f(x) = 2x + 5 \)[/tex].
[tex]\[
f(-2) = 2(-2) + 5 = -4 + 5 = 1
\][/tex]
### Finding [tex]\( f(0) \)[/tex]
- The value 0 fits in the condition [tex]\(-1 < 0 \leq 3\)[/tex], so we use the formula [tex]\( f(x) = 4x^2 + 2 \)[/tex].
[tex]\[
f(0) = 4(0)^2 + 2 = 0 + 2 = 2
\][/tex]
### Finding [tex]\( f(4) \)[/tex]
- Since [tex]\( 4 > 3 \)[/tex], we use the formula [tex]\( f(x) = 6x^2 - 2 \)[/tex].
[tex]\[
f(4) = 6(4)^2 - 2 = 6(16) - 2 = 96 - 2 = 94
\][/tex]
In conclusion, the values are:
- [tex]\( f(-2) = 1 \)[/tex]
- [tex]\( f(0) = 2 \)[/tex]
- [tex]\( f(4) = 94 \)[/tex]
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