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Suppose that the function [tex] f [/tex] is defined for all real numbers as follows:

[tex]\ \textless \ br/\ \textgreater \ f(x) = \begin{cases}\ \textless \ br/\ \textgreater \ \frac{1}{2} x^2 - 5 & \text{if } x \neq 2 \\\ \textless \ br/\ \textgreater \ 3 & \text{if } x = 2\ \textless \ br/\ \textgreater \ \end{cases}\ \textless \ br/\ \textgreater \ [/tex]

Find [tex] f(-1) [/tex], [tex] f(2) [/tex], and [tex] f(4) [/tex].

[tex]\ \textless \ br/\ \textgreater \ \begin{array}{c}\ \textless \ br/\ \textgreater \ f(-1) = \\\ \textless \ br/\ \textgreater \ f(2) = \\\ \textless \ br/\ \textgreater \ f(4) = \ \textless \ br/\ \textgreater \ \end{array}\ \textless \ br/\ \textgreater \ [/tex]



Answer :

GinnyAnswer
To determine [tex]\(f(-1)\)[/tex], [tex]\(f(2)\)[/tex], and [tex]\(f(4)\)[/tex] for the given function [tex]\(f(x)\)[/tex], we can evaluate the function based on the given piecewise definition. Let's go through each value one by one.

1. Finding [tex]\(f(-1)\)[/tex]:
- For [tex]\(x = -1\)[/tex], the function definition states [tex]\(f(x) = \frac{1}{2} x^2 - 5\)[/tex] if [tex]\(x \neq 2\)[/tex].
- Substitute [tex]\(x = -1\)[/tex] into the function:
[tex]\[ f(-1) = \frac{1}{2} (-1)^2 - 5 = \frac{1}{2} \cdot 1 - 5 = 0.5 - 5 = -4.5 \][/tex]

2. Finding [tex]\(f(2)\)[/tex]:
- For [tex]\(x = 2\)[/tex], the function definition directly gives the value:
[tex]\[ f(2) = 3 \][/tex]

3. Finding [tex]\(f(4)\)[/tex]:
- For [tex]\(x = 4\)[/tex], the function definition states [tex]\(f(x) = \frac{1}{2} x^2 - 5\)[/tex] if [tex]\(x \neq 2\)[/tex].
- Substitute [tex]\(x = 4\)[/tex] into the function:
[tex]\[ f(4) = \frac{1}{2} \cdot 4^2 - 5 = \frac{1}{2} \cdot 16 - 5 = 8 - 5 = 3 \][/tex]

So, the values are:
[tex]\[ f(-1) = -4.5 \][/tex]
[tex]\[ f(2) = 3 \][/tex]
[tex]\[ f(4) = 3 \][/tex]

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