Answer :
To find the values of the function [tex]\( f \)[/tex] at the given points, we need to evaluate the piecewise function accordingly. The function [tex]\( f \)[/tex] is defined as follows:
[tex]\[ f(x) = \begin{cases} \frac{1}{3} x^2 - 4 & \text{ if } x \neq -2 \\ 2 & \text{ if } x = -2 \end{cases} \][/tex]
### Finding [tex]\( f(-5) \)[/tex]:
Since [tex]\( -5 \neq -2 \)[/tex], we use the first piece of the function.
[tex]\[ f(-5) = \frac{1}{3} (-5)^2 - 4 \][/tex]
Calculating [tex]\( (-5)^2 \)[/tex]:
[tex]\[ (-5)^2 = 25 \][/tex]
Substituting this back into the equation:
[tex]\[ f(-5) = \frac{1}{3} \cdot 25 - 4 \][/tex]
[tex]\[ f(-5) = \frac{25}{3} - 4 \][/tex]
Converting 4 to a fraction with a denominator of 3:
[tex]\[ 4 = \frac{12}{3} \][/tex]
Subtracting the fractions:
[tex]\[ f(-5) = \frac{25}{3} - \frac{12}{3} \][/tex]
[tex]\[ f(-5) = \frac{13}{3} \][/tex]
In decimal form:
[tex]\[ f(-5) \approx 4.333333333333332 \][/tex]
### Finding [tex]\( f(-2) \)[/tex]:
Since [tex]\( -2 = -2 \)[/tex], we use the second piece of the function.
[tex]\[ f(-2) = 2 \][/tex]
### Finding [tex]\( f(5) \)[/tex]:
Since [tex]\( 5 \neq -2 \)[/tex], we use the first piece of the function.
[tex]\[ f(5) = \frac{1}{3} (5)^2 - 4 \][/tex]
Calculating [tex]\( (5)^2 \)[/tex]:
[tex]\[ (5)^2 = 25 \][/tex]
Substituting this back into the equation:
[tex]\[ f(5) = \frac{1}{3} \cdot 25 - 4 \][/tex]
[tex]\[ f(5) = \frac{25}{3} - 4 \][/tex]
Converting 4 to a fraction with a denominator of 3:
[tex]\[ 4 = \frac{12}{3} \][/tex]
Subtracting the fractions:
[tex]\[ f(5) = \frac{25}{3} - \frac{12}{3} \][/tex]
[tex]\[ f(5) = \frac{13}{3} \][/tex]
In decimal form:
[tex]\[ f(5) \approx 4.333333333333332 \][/tex]
### Summary:
Thus, the values of the function at the given points are:
[tex]\[ f(-5) = 4.333333333333332 \][/tex]
[tex]\[ f(-2) = 2 \][/tex]
[tex]\[ f(5) = 4.333333333333332 \][/tex]
Therefore:
[tex]\[ \begin{array}{c} f(-5)= 4.333333333333332 \\ f(-2)= 2 \\ f(5)= 4.333333333333332 \end{array} \][/tex]
[tex]\[ f(x) = \begin{cases} \frac{1}{3} x^2 - 4 & \text{ if } x \neq -2 \\ 2 & \text{ if } x = -2 \end{cases} \][/tex]
### Finding [tex]\( f(-5) \)[/tex]:
Since [tex]\( -5 \neq -2 \)[/tex], we use the first piece of the function.
[tex]\[ f(-5) = \frac{1}{3} (-5)^2 - 4 \][/tex]
Calculating [tex]\( (-5)^2 \)[/tex]:
[tex]\[ (-5)^2 = 25 \][/tex]
Substituting this back into the equation:
[tex]\[ f(-5) = \frac{1}{3} \cdot 25 - 4 \][/tex]
[tex]\[ f(-5) = \frac{25}{3} - 4 \][/tex]
Converting 4 to a fraction with a denominator of 3:
[tex]\[ 4 = \frac{12}{3} \][/tex]
Subtracting the fractions:
[tex]\[ f(-5) = \frac{25}{3} - \frac{12}{3} \][/tex]
[tex]\[ f(-5) = \frac{13}{3} \][/tex]
In decimal form:
[tex]\[ f(-5) \approx 4.333333333333332 \][/tex]
### Finding [tex]\( f(-2) \)[/tex]:
Since [tex]\( -2 = -2 \)[/tex], we use the second piece of the function.
[tex]\[ f(-2) = 2 \][/tex]
### Finding [tex]\( f(5) \)[/tex]:
Since [tex]\( 5 \neq -2 \)[/tex], we use the first piece of the function.
[tex]\[ f(5) = \frac{1}{3} (5)^2 - 4 \][/tex]
Calculating [tex]\( (5)^2 \)[/tex]:
[tex]\[ (5)^2 = 25 \][/tex]
Substituting this back into the equation:
[tex]\[ f(5) = \frac{1}{3} \cdot 25 - 4 \][/tex]
[tex]\[ f(5) = \frac{25}{3} - 4 \][/tex]
Converting 4 to a fraction with a denominator of 3:
[tex]\[ 4 = \frac{12}{3} \][/tex]
Subtracting the fractions:
[tex]\[ f(5) = \frac{25}{3} - \frac{12}{3} \][/tex]
[tex]\[ f(5) = \frac{13}{3} \][/tex]
In decimal form:
[tex]\[ f(5) \approx 4.333333333333332 \][/tex]
### Summary:
Thus, the values of the function at the given points are:
[tex]\[ f(-5) = 4.333333333333332 \][/tex]
[tex]\[ f(-2) = 2 \][/tex]
[tex]\[ f(5) = 4.333333333333332 \][/tex]
Therefore:
[tex]\[ \begin{array}{c} f(-5)= 4.333333333333332 \\ f(-2)= 2 \\ f(5)= 4.333333333333332 \end{array} \][/tex]
