What is the long-run behavior of the functions in the following problem?

$$y=\frac{2^x+3}{x^2+5}$$

A function f satisfies f(1)=2 and f(3)=18. Find f(2) if

(a) f is linear.

(b) f is exponential.

(c) f is a power function.

For the polynomials in the following exercise, state the degree and the number of terms, and describe the long-run behavior.

$$y=2 x^3-3 x+7$$

In this problem, give a possible formula for the polynomial.

(-1,0), (-1/2, -27/16), (0,0), (1,0)

The table below gives approximate values for three functions, f, g, and h. One is exponential, one is trigonometric, and one is a power function. Determine which is which and find possible formulas for each.

An alcohol solution consists of 5 gallons of pure water and x gallons of alcohol, x>0. Let f(x) be the ratio of the volume of alcohol to the total volume of liquid. [Note that $f(x)$ is the concentration of the alcohol in the solution.]

(a) Find a possible formula for $f(x)$.

(b) Evaluate and interpret $f(7)$ in the context of the mixture.

(c) What is the zero of $f$ ? Interpret your result in the context of the mixture.

(d) Find an equation for the horizontal asymptote of $f$. Explain its significance in the context of the mixture.

The given figure shows the cost function, C(n), from the previous problem, and a line, $l$, that passes through the origin.

(a) What is the slope of line $l$?

(b) How does line $l$ relate to $a\left(n_0\right)$, the average cost of producing $n_0$ units?

Anthropologists suggest that the relationship between the body weight and brain weight of primates can be modeled with a power function. The table below lists various body weights and the corresponding brain weights of different primates.

(a) Using the table below, find a power function that gives the brain weight, $Q$ (in $\mathrm{mg}$ ), as a function of the body weight, $b$ (in gm).

(b) The erythrocebus (Patas monkey) has a body weight of $7800 \mathrm{gm}$. Estimate its brain weight.

In calculus, we often consider pairs of polynomials. If $F(x)=3 x^4-4 x^3+5 x-4$, find $a_4, a_3, a_2, a_1, a_0$. Then use these values to construct the cubic polynomial

$$f(x)=4 a_4 x^3+3 a_3 x^2+2 a_2 x+a_1 .$$

Can the formula in the following exercise be written in the form of an exponential function or a power function? If not, explain why the function does not fit either form.

$$q(x)=5^{\left(x^2\right)}$$

The following problem give value of transformations of either y= 1 / x or y=1 / x$^2$. In each case

(a) Determine if the values are from a transformation of $y=1 / x$ or $y=1 / x^2$. Explain your reasoning.

(b) Find a possible formula for the function.

For the following problem, find the real zeros (if any) of the polynomials.

$$y=4 x^2-1$$

Describe the behavior of y=x$^{-3}$ and y=x$^{1 / 3}$ as

(a) $x \rightarrow 0$ from the right

(b) $x \rightarrow \infty$

For the following exercise, perform the operations. Express answers in reduced form.

$$\dfrac{1}{2 x}-\dfrac{2}{3}$$