Question

10 points, please help me and explain how to do this with answers!

Answer 1

$\bf f(x)=log\left( \cfrac{x}{8} \right)\\\\ -----------------------------\\\\ \textit{x-intercept, setting f(x)=0} \\\\ 0=log\left( \cfrac{x}{8} \right)\implies 0=log(x)-log(8)\implies log(8)=log(x) \\\\ 8=x\\\\ -----------------------------$

$\bf \textit{y-intercept, is setting x=0}\\ \textit{wait just a second!, a logarithm never gives 0} \\\\ log_{{ a}}{{ b}}=y \iff {{ a}}^y={{ b}}\qquad\qquad % exponential notation 2nd form {{ a}}^y={{ b}}\iff log_{{ a}}{{ b}}=y \\\\ \textit{now, what exponent for "a" can give you a zero? none}\\ \textit{so, there's no y-intercept, because "x" is never 0 in }\frac{x}{8}\\ \textit{that will make the fraction to 0, and a}\\ \textit{logarithm will never give that, 0 or a negative}$

$\bf -----------------------------\\\\ domain \\\\ \textit{since whatever value "x" is, cannot make the fraction}\\ \textit{negative or become 0, , then the domain is }x\ \textgreater \ 0\\\\ -----------------------------\\\\ range \\\\ \textit{those values for "x", will spit out, pretty much}\\ \textit{any "y", including negative exponents, thus}\\ \textit{range is }(-\infty,+\infty)$
 p, li { white-space: pre-wrap; }

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now on 2)

$\bf f(x)=\cfrac{3}{x^4}$   if the denominator has a higher degree than the numerator, the horizontal asymptote is y = 0, or the x-axis,

in this case, the numerator has a degree of 0, the denominator has 4, thus y = 0

vertical asymptotes occur when the denominator is 0, that is, when the fraction becomes undefined, and for this one, that occurs at  $x^4=0\implies x=0$  or the y-axis

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now on 3)

$\bf f(x)=\cfrac{1}{x}$

now, let's see some transformations templates

$\bf \qquad \qquad \qquad \qquad \textit{function transformations} \\ \quad \\ \begin{array}{rllll} % left side templates f(x)=&{{ A}}({{ B}}x+{{ C}})+{{ D}} \\ \quad \\ y=&{{ A}}({{ B}}x+{{ C}})+{{ D}} \\ \quad \\ f(x)=&{{ A}}\sqrt{{{ B}}x+{{ C}}}+{{ D}} \\ \quad \\ f(x)=&{{ A}}\mathbb{R}^{{{ B}}x+{{ C}}}+{{ D}} \end{array}$

$\bf \begin{array}{llll} % right side info \bullet \textit{ stretches or shrinks horizontally by } {{ A}}\cdot {{ B}}\\ \bullet \textit{ horizontal shift by }\frac{{{ C}}}{{{ B}}}\\ \qquad if\ \frac{{{ C}}}{{{ B}}}\textit{ is negative, to the right}\\ \qquad if\ \frac{{{ C}}}{{{ B}}}\textit{ is positive, to the left}\\ \bullet \textit{ vertical shift by }{{ D}}\\ \qquad if\ {{ D}}\textit{ is negative, downwards}\\ \qquad if\ {{ D}}\textit{ is positive, upwards} \end{array}$

now, let's take a peek at g(x)

$\bf \begin{array}{lcllll} g(x)=&-&\cfrac{1}{x}&+3\\ &\uparrow &&\uparrow \\ &\textit{upside down}&& \begin{array}{llll} \textit{vertical shift up}\\ \textit{by 3 units} \end{array} \end{array}$