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)

Question

Accepted Answer

∻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 $x$.⚹◊What is the domain of the function $f(x) = x^2 - x - 5$?⍭ Generate me a similar question◊

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