Knowledge Check: Chapters 1–6

For$f(x)=-2x^2+3x$, evaluate$f(-3)$.

Find$(f\circ g)(x)$for$f(x)=3x+2$and$g(x)=5-6x$.

Using the same functions, find$(g\circ f)(x)$.

Solve$\lvert x+4\rvert=18$. Enter both solutions, separated by a comma.

Find$f^{-1}(x)$for$f(x)=9+10x$.

If$f(5)=2$for a one-to-one function$f$, find$f^{-1}(2)$.

Find a linear equation, in slope-intercept form, for the line that passes through$(7,5)$and$(3,17)$.

Write an equation, in slope-intercept form, for the line perpendicular to$f(x)=5x-1$that passes through the point$(5,20)$.

A town’s population increases at a constant rate. In 2010 the population was 55,000. By 2012 the population had increased to 76,000. If this trend continues, predict the population in 2016.

The median home value in the Pima Central subdivision (adjusted for inflation) was $32,000 in 1970 and $85,000 in 2010. Assume the value changed linearly over that period. If this trend continues, what would be the median home value in Pima Central in 2015, in dollars?

Perform the indicated operation and express the result as a simplified complex number:$(4+3i)+(-2-5i)$.

Solve$x^2-4x+5=0$over the complex number system. Enter both solutions, separated by a comma.

A rectangular plot of land is to be enclosed by fencing. One side is along a river and so needs no fence. If the total fencing available is 600 meters, find the length of the side parallel to the river that gives the plot maximum area.

A rectangular plot of land is to be enclosed by fencing. One side is along a river and so needs no fence. If the total fencing available is 600 meters, find the width of the plot (the length of each side perpendicular to the river) that gives the plot maximum area.

Identify the degree of the polynomial function$f(x)=4x^5-3x^3+2x-1$.

Identify the leading coefficient of that same polynomial function,$f(x)=4x^5-3x^3+2x-1$.

Find all zeros of$f(x)=(x+3)^2(2x-1)(x+1)^3$. Enter all of them, separated by commas.

What is the multiplicity of the zero$x=-1$for that same function,$f(x)=(x+3)^2(2x-1)(x+1)^3$?

Divide$x^3-2x^2+4x+4$by$x-2$using long division. Give the quotient.

Give the remainder from that same division of$x^3-2x^2+4x+4$by$x-2$.

Use the Rational Zero Theorem to solve$2x^3-3x^2-18x-8=0$. Enter all solutions, separated by commas.

Find the vertical asymptote of$f(x)=\tfrac{x+2}{x-5}$.

Find the horizontal asymptote of that same function,$f(x)=\tfrac{x+2}{x-5}$.

Find the inverse of$f(x)=(x-2)^2$, given that its domain is restricted to$x\ge2$.

Find the inverse of$f(x)=\sqrt{4x+5}-3$.

$y$varies directly as the square of$x$. If$x=3$gives$y=36$, find$y$when$x=4$.

$y$varies jointly as the cube of$x$and as$z$. If$x=1$and$z=2$give$y=6$, find$y$when$x=2$and$z=3$.

Find an exponential equation that passes through the points$(2,2.25)$and$(5,60.75)$.

What is the range of$f(x)=3.5(2)^x$? Write your answer in interval notation.

The graph of$f(x)=6.5^x$is reflected about the$y$-axis and stretched vertically by a factor of$7$. What is the equation of the new function,$g(x)$?

Solve for$x$:$\log_{64}(x)=\tfrac{1}{3}$.

Evaluate$\log(0.000001)$without using a calculator.

What is the domain of$g(x)=\ln(4x+20)-17$? Write your answer in interval notation.

What is the vertical asymptote of that same function,$g(x)=\ln(4x+20)-17$?

Rewrite$\log_8(x)+\log_8(5)+\log_8(y)+\log_8(13)$in compact form.

Use properties of logarithms to expand$\ln\left(2b\sqrt{\tfrac{b+1}{b-1}}\right)$.

Solve$\tfrac{125}{\left(\tfrac{1}{625}\right)^{-x-3}}=5^3$for$x$.

Find the exact solution for$e^{2x}-e^x-110=0$.

The equation$N(t)=\frac{1{,}200}{1+199e^{-0.625t}}$models the number of people in a school who have heard a rumor after$t$days. To the nearest tenth, how many days will it be before the rumor spreads to half its carrying capacity?

A doctor prescribes$300$milligrams of a therapeutic drug that decays by about$17\%$each hour. To the nearest hundredth of a milligram, how much of the drug will remain in the patient’s system after$24$hours?

The population of a culture of bacteria is modeled by the logistic equation$P(t)=\frac{14{,}250}{1+29e^{-0.62t}}$, where$t$is in days. To the nearest tenth, how many days will it take the culture to reach$75\%$of its carrying capacity?

Convert$-210^\circ$to radians.

Find the length of an arc in a circle of radius 7 meters subtended by a central angle of$85^\circ$. Round your answer to three decimal places.

Find the exact value of$\sin\left(\tfrac{\pi}{3}\right)$.

State the reference angle for$\tfrac{3\pi}{4}$.

Find the exact value of$\tan\left(\tfrac{\pi}{4}\right)$.

Use reference angles to evaluate$\sec(315^\circ)$.

Fill in the missing angle, in degrees:$\cos\left(\tfrac{\pi}{2}\right)=\sin\left(\underline{\qquad}\right)$.

A 15-ft ladder leans against a building so that the angle between the ground and the ladder is$70^\circ$. How high does the ladder reach up the side of the building? Round to four decimal places.

Find the period of$y=\sin\left(\tfrac{\pi}{6}x+\pi\right)-3$.

Determine the phase shift for$y=\sin\left(\tfrac{\pi}{6}x+\pi\right)-3$as a signed value (positive = right, negative = left).

What is the period of$g(x)=3\tan(6x+42)$?

What is the horizontal shift of$g(x)=3\tan(6x+42)$?

Find the exact value of$\cos^{-1}\left(\tfrac{\sqrt3}{2}\right)$.

Find the exact value of$\tan\left(\cos^{-1}\left(\tfrac{5}{13}\right)\right)$.