12. Use transformations of the parent graph of $m(x)=\llbracket x \rrbracket$ to sketch the graph of $p(x)=2 \llbracket x-3 \rrbracket$.
13. Describe the transformations relating the graph of $g(x)$ $=\frac{1}{2}|-6 x-3|+4$ to the graph of its parent function $f(x)=|x|$.
14. Graph $f(x)=\frac{1}{|x|}-1$.
15. If $f(x)=2 x-1$ and $g(x)=\frac{1}{2 x^{2}}$, find $\left(\frac{f}{g}\right)(x)$ and its domain.
16. CONES A cone with a fixed height of 81 millimeters is shown on a computer screen. An animator increases the radius $r$ at a rate of 4.5 centimeters per minute. Write the function that gives the volume $v(r)$ of the cone in cubic centimeters as a function of time $f(t)$. Assume the radius is 4.5 centimeters at $t=1$.
17. If $f(x)=3 x^{2}+4$ and $g(x)=\frac{1}{x^{2}-x}$, find $[g \circ f](x)$.
18. Find the inverse of $f(x)=\frac{3}{x-2}$.
19. Determine if $f(x)=\frac{1}{2 x^{2}}$ is a one-to-one function.
20. CONSTRUCTION A construction worker orders 50 boxes of screws. Some are wood screws at $\$ 2.93 /$ box and some are sheet metal screws at $\$ 4.07 /$ box. Write the function that can be used to find the number of boxes of wood screws ordered if given the total price paid.

Bonus Given the graph of $p(x)$, sketch the graph of
$$
y=-2 p\left[\frac{1}{2}(x-2)\right]+2
$$

Question

12. Use transformations of the parent graph of $m(x)=\llbracket x \rrbracket$ to sketch the graph of $p(x)=2 \llbracket x-3 \rrbracket$.
13. Describe the transformations relating the graph of $g(x)$ $=\frac{1}{2}|-6 x-3|+4$ to the graph of its parent function $f(x)=|x|$.
14. Graph $f(x)=\frac{1}{|x|}-1$.
15. If $f(x)=2 x-1$ and $g(x)=\frac{1}{2 x^{2}}$, find $\left(\frac{f}{g}\right)(x)$ and its domain.
16. CONES A cone with a fixed height of 81 millimeters is shown on a computer screen. An animator increases the radius $r$ at a rate of 4.5 centimeters per minute. Write the function that gives the volume $v(r)$ of the cone in cubic centimeters as a function of time $f(t)$. Assume the radius is 4.5 centimeters at $t=1$.
17. If $f(x)=3 x^{2}+4$ and $g(x)=\frac{1}{x^{2}-x}$, find $[g \circ f](x)$.
18. Find the inverse of $f(x)=\frac{3}{x-2}$.
19. Determine if $f(x)=\frac{1}{2 x^{2}}$ is a one-to-one function.
20. CONSTRUCTION A construction worker orders 50 boxes of screws. Some are wood screws at $\$ 2.93 /$ box and some are sheet metal screws at $\$ 4.07 /$ box. Write the function that can be used to find the number of boxes of wood screws ordered if given the total price paid.

Bonus Given the graph of $p(x)$, sketch the graph of
$$
y=-2 p\left[\frac{1}{2}(x-2)\right]+2
$$

Accepted Answer

∻Solution by Steps∻ ### Question 12 **Use transformations of the parent graph of $m(x)=\llbracket x rbracket$ to sketch the graph of $p(x)=2 \llbracket x-3 rbracket$.** ‖ step 1 ⋮ The parent function is $m(x)=\llbracket x rbracket$, which is the floor function. ‖ ‖ step 2 ⋮ The transformation $x-3$ inside the floor function shifts the graph 3 units to the right. ‖ ‖ step 3 ⋮ The multiplication by 2 outside the floor function scales the graph vertically by a factor of 2. ‖ ‖ step 4 ⋮ Therefore, $p(x)=2 \llbracket x-3 rbracket$ is the graph of $m(x)=\llbracket x rbracket$ shifted 3 units to the right and stretched vertically by a factor of 2. ‖ ∻Answer∻ ⚹ The graph of $p(x)=2 \llbracket x-3 rbracket$ is the graph of $m(x)=\llbracket x rbracket$ shifted 3 units to the right and stretched vertically by a factor of 2. ⚹ ∻Key Concept∻ ⚹Transformations of the floor function⚹ ∻Explanation∻ ⚹Shifting and scaling transformations applied to the floor function result in the graph of $p(x)$.⚹ ### Question 13 **Describe the transformations relating the graph of $g(x)=\frac{1}{2}|-6 x-3|+4$ to the graph of its parent function $f(x)=|x|$.** ‖ step 1 ⋮ The parent function is $f(x)=|x|$. ‖ ‖ step 2 ⋮ The transformation $-6x$ inside the absolute value reflects the graph across the y-axis and scales it horizontally by a factor of $\frac{1}{6}$. ‖ ‖ step 3 ⋮ The transformation $-3$ inside the absolute value shifts the graph 3 units to the left. ‖ ‖ step 4 ⋮ The multiplication by $\frac{1}{2}$ outside the absolute value scales the graph vertically by a factor of $\frac{1}{2}$. ‖ ‖ step 5 ⋮ The addition of 4 outside the absolute value shifts the graph 4 units up. ‖ ∻Answer∻ ⚹ The graph of $g(x)=\frac{1}{2}|-6 x-3|+4$ is the graph of $f(x)=|x|$ reflected across the y-axis, scaled horizontally by $\frac{1}{6}$, shifted 3 units to the left, scaled vertically by $\frac{1}{2}$, and shifted 4 units up. ⚹ ∻Key Concept∻ ⚹Transformations of the absolute value function⚹ ∻Explanation∻ ⚹Reflections, scalings, and shifts applied to the absolute value function result in the graph of $g(x)$.⚹ ### Question 14 **Graph $f(x)=\frac{1}{|x|}-1$.** ‖ step 1 ⋮ The parent function is $f(x)=\frac{1}{|x|}$. ‖ ‖ step 2 ⋮ The transformation $-1$ outside the function shifts the graph 1 unit down. ‖ ‖ step 3 ⋮ The graph of $f(x)=\frac{1}{|x|}-1$ has vertical asymptotes at $x=0$ and horizontal asymptote at $y=-1$. ‖ ‖ step 4 ⋮ The graph is symmetric with respect to the y-axis. ‖ ∻Answer∻ ⚹ The graph of $f(x)=\frac{1}{|x|}-1$ has vertical asymptotes at $x=0$, a horizontal asymptote at $y=-1$, and is symmetric with respect to the y-axis. ⚹ ∻Key Concept∻ ⚹Graphing rational functions with absolute values⚹ ∻Explanation∻ ⚹Shifting the graph of $\frac{1}{|x|}$ down by 1 unit results in the graph of $f(x)$.⚹ ### Question 15 **If $f(x)=2 x-1$ and $g(x)=\frac{1}{2 x^{2}}$, find $\left(\frac{f}{g} ight)(x)$ and its domain.** ‖ step 1 ⋮ The functions are $f(x)=2 x-1$ and $g(x)=\frac{1}{2 x^{2}}$. ‖ ‖ step 2 ⋮ The quotient function is $\left(\frac{f}{g} ight)(x)=\frac{f(x)}{g(x)}=\frac{2 x-1}{\frac{1}{2 x^{2}}}$. ‖ ‖ step 3 ⋮ Simplify the quotient: $\left(\frac{f}{g} ight)(x)=(2 x-1) \cdot (2 x^{2})=4 x^{3}-2 x^{2}$. ‖ ‖ step 4 ⋮ The domain of $\left(\frac{f}{g} ight)(x)$ is all real numbers except $x=0$ (where $g(x)$ is undefined). ‖ ∻Answer∻ ⚹ $\left(\frac{f}{g} ight)(x)=4 x^{3}-2 x^{2}$ with domain $x eq 0$. ⚹ ∻Key Concept∻ ⚹Quotient of functions⚹ ∻Explanation∻ ⚹The quotient of two functions is found by dividing the functions and simplifying, with the domain excluding points where the denominator is zero.⚹ ### Question 16 **A cone with a fixed height of 81 millimeters is shown on a computer screen. An animator increases the radius $r$ at a rate of 4.5 centimeters per minute. Write the function that gives the volume $v(r)$ of the cone in cubic centimeters as a function of time $f(t)$. Assume the radius is 4.5 centimeters at $t=1$.** ‖ step 1 ⋮ The volume of a cone is given by $V=\frac{1}{3}\pi r^{2}h$. ‖ ‖ step 2 ⋮ Given $h=81$ mm = 8.1 cm, the volume function is $V(r)=\frac{1}{3}\pi r^{2}(8.1)=2.7\pi r^{2}$. ‖ ‖ step 3 ⋮ The radius $r$ increases at a rate of 4.5 cm/min, so $r(t)=4.5t$. ‖ ‖ step 4 ⋮ Substitute $r(t)$ into $V(r)$: $V(t)=2.7\pi (4.5t)^{2}=2.7\pi (20.25t^{2})=54.675\pi t^{2}$. ‖ ∻Answer∻ ⚹ The volume function is $V(t)=54.675\pi t^{2}$ cubic centimeters. ⚹ ∻Key Concept∻ ⚹Volume of a cone as a function of time⚹ ∻Explanation∻ ⚹The volume of a cone can be expressed as a function of time by substituting the time-dependent radius into the volume formula.⚹ ### Question 17 **If $f(x)=3 x^{2}+4$ and $g(x)=\frac{1}{x^{2}-x}$, find $[g \circ f](x)$.** ‖ step 1 ⋮ The functions are $f(x)=3 x^{2}+4$ and $g(x)=\frac{1}{x^{2}-x}$. ‖ ‖ step 2 ⋮ The composition $[g \circ f](x)=g(f(x))=g(3 x^{2}+4)$. ‖ ‖ step 3 ⋮ Substitute $f(x)$ into $g(x)$: $g(3 x^{2}+4)=\frac{1}{(3 x^{2}+4)^{2}-(3 x^{2}+4)}$. ‖ ‖ step 4 ⋮ Simplify the expression: $g(3 x^{2}+4)=\frac{1}{9 x^{4}+24 x^{2}+16-3 x^{2}-4}=\frac{1}{9 x^{4}+21 x^{2}+12}$. ‖ ∻Answer∻ ⚹ $[g \circ f](x)=\frac{1}{9 x^{4}+21 x^{2}+12}$. ⚹ ∻Key Concept∻ ⚹Composition of functions⚹ ∻Explanation∻ ⚹The composition of two functions involves substituting one function into the other and simplifying the result.⚹ ### Question 18 **Find the inverse of $f(x)=\frac{3}{x-2}$.** ‖ step 1 ⋮ The function is $f(x)=\frac{3}{x-2}$. ‖ ‖ step 2 ⋮ To find the inverse, swap $x$ and $y$: $x=\frac{3}{y-2}$. ‖ ‖ step 3 ⋮ Solve for $y$: $x(y-2)=3 \Rightarrow y-2=\frac{3}{x} \Rightarrow y=\frac{3}{x}+2$. ‖ ‖ step 4 ⋮ The inverse function is $f^{-1}(x)=\frac{3}{x}+2$. ‖ ∻Answer∻ ⚹ The inverse function is $f^{-1}(x)=\frac{3}{x}+2$. ⚹ ∻Key Concept∻ ⚹Finding the inverse of a function⚹ ∻Explanation∻ ⚹To find the inverse of a function, swap the variables and solve for the new dependent variable.⚹ ### Question 19 **Determine if $f(x)=\frac{1}{2 x^{2}}$ is a one-to-one function.** ‖ step 1 ⋮ The function is $f(x)=\frac{1}{2 x^{2}}$. ‖ ‖ step 2 ⋮ A function is one-to-one if each $y$ value corresponds to exactly one $x$ value. ‖ ‖ step 3 ⋮ $f(x)=\frac{1}{2 x^{2}}$ is not one-to-one because $f(x)=f(-x)$. ‖ ‖ step 4 ⋮ Therefore, $f(x)=\frac{1}{2 x^{2}}$ is not injective (one-to-one). ‖ ∻Answer∻ ⚹ $f(x)=\frac{1}{2 x^{2}}$ is not a one-to-one function. ⚹ ∻Key Concept∻ ⚹One-to-one functions⚹ ∻Explanation∻ ⚹A function is one-to-one if each output is associated with a unique input.⚹ ### Question 20 **A construction worker orders 50 boxes of screws. Some are wood screws at \$2.93 per box and some are sheet metal screws at \$4.07 per box. Write the function that can be used to find the number of boxes of wood screws ordered if given the total price paid.** ‖ step 1 ⋮ Let $x$ be the number of boxes of wood screws. ‖ ‖ step 2 ⋮ The number of boxes of sheet metal screws is $50-x$. ‖ ‖ step 3 ⋮ The total price is given by $2.93x + 4.07(50-x)$. ‖ ‖ step 4 ⋮ Simplify the expression: $2.93x + 4.07(50-x) = 2.93x + 203.5 - 4.07x = 203.5 - 1.14x$. ‖ ‖ step 5 ⋮ The function for the total price is $P(x)=203.5 - 1.14x$. ‖ ∻Answer∻ ⚹ The function for the total price is $P(x)=203.5 - 1.14x$. ⚹ ∻Key Concept∻ ⚹Linear functions in cost problems⚹ ∻Explanation∻ ⚹The total cost can be expressed as a linear function of the number of boxes of wood screws.⚹ ### Bonus Question **Given the graph of $p(x)$, sketch the graph of $y=-2 p\left[\frac{1}{2}(x-2) ight]+2$.** ‖ step 1 ⋮ The given transformation involves $p(x)$. ‖ ‖ step 2 ⋮ The transformation $\frac{1}{2}(x-2)$ inside the function scales the graph horizontally by a factor of 2 and shifts it 2 units to the right. ‖ ‖ step 3 ⋮ The multiplication by $-2$ outside the function reflects the graph across the x-axis and scales it vertically by a factor of 2. ‖ ‖ step 4 ⋮ The addition of 2 outside the function shifts the graph 2 units up. ‖ ∻Answer∻ ⚹ The graph of $y=-2 p\left[\frac{1}{2}(x-2) ight]+2$ is the graph of $p(x)$ scaled horizontally by 2, shifted 2 units to the right, reflected across the x-axis, scaled vertically by 2, and shifted 2 units up. ⚹ ∻Key Concept∻ ⚹Transformations of functions⚹ ∻Explanation∻ ⚹Combining horizontal and vertical transformations results in the new graph.⚹◊What is the rate of change of the radius in the volume function of the cone mentioned⍭ Generate me a similar question◊

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