Question

Math

Posted 3 months ago

```
Let $f$ be the function defined by the rule $f(x)=x^{2}-x-5$.
(a) Find the domain of $f$. (Enter your answer using interval notation.)
(b) Compute $f(x)$ for $x=-3,-2,-1,0, \frac{1}{2}, 1,2,3$.
$\begin{aligned}
f(-3) & = \\
f(-2) & = \\
f(-1) & = \\
f(0) & = \\
f\left(\frac{1}{2}\right) & = \\
f(1) & = \\
f(2) & = \\
f(3) & =
\end{aligned}$
$\square$
$\square$
$\square$
$\square$
$\square$
$\square$
$\square$
$\square$
(c) Use the results obtained in parts (a) and (b) to sketch the graph of $f$.
(i)
```

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Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

To find the domain of the function $f(x) = x^2 - x - 5$, we note that it is a polynomial function. Polynomial functions are defined for all real numbers. Therefore, the domain of $f$ is all real numbers

step 2

To compute $f(x)$ for $x = -3, -2, -1, 0, \frac{1}{2}, 1, 2, 3$, we substitute each value of $x$ into the function $f(x) = x^2 - x - 5$

step 3

For $x = -3$:
$f(-3) = (-3)^2 - (-3) - 5 = 9 + 3 - 5 = 7$

step 4

For $x = -2$:
$f(-2) = (-2)^2 - (-2) - 5 = 4 + 2 - 5 = 1$

step 5

For $x = -1$:
$f(-1) = (-1)^2 - (-1) - 5 = 1 + 1 - 5 = -3$

step 6

For $x = 0$:
$f(0) = 0^2 - 0 - 5 = -5$

step 7

For $x = \frac{1}{2}$:
$f\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^2 - \frac{1}{2} - 5 = \frac{1}{4} - \frac{1}{2} - 5 = -\frac{21}{4}$

step 8

For $x = 1$:
$f(1) = 1^2 - 1 - 5 = 1 - 1 - 5 = -5$

step 9

For $x = 2$:
$f(2) = 2^2 - 2 - 5 = 4 - 2 - 5 = -3$

step 10

For $x = 3$:
$f(3) = 3^2 - 3 - 5 = 9 - 3 - 5 = 1$

step 11

Using the results from steps 3 to 10, we can sketch the graph of $f(x) = x^2 - x - 5$. The points to plot are: $(-3, 7)$, $(-2, 1)$, $(-1, -3)$, $(0, -5)$, $\left(\frac{1}{2}, -\frac{21}{4}\right)$, $(1, -5)$, $(2, -3)$, $(3, 1)$

Answer

The domain of $f$ is all real numbers. The computed values are $f(-3) = 7$, $f(-2) = 1$, $f(-1) = -3$, $f(0) = -5$, $f\left(\frac{1}{2}\right) = -\frac{21}{4}$, $f(1) = -5$, $f(2) = -3$, $f(3) = 1$. The graph can be sketched using these points.

Key Concept

Domain and range of polynomial functions

Explanation

Polynomial functions are defined for all real numbers, and their graphs can be sketched by plotting points obtained from evaluating the function at specific values of $x$.

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