Question

1. Determine the equation of the cubic function of the family with y-intercept 5.

2. Determine the equation of the cubic function of the family that passes through the point (3, –24).

3. Solve the equation 6cos^2x +7cosx-5=0, for -2piπ ≤ x ≤ 2pi. Express answer as an exact value.

4. The pH of a solution is calculated by using the formula pH=-log[H^+], where [H^+]is the concentration of the hydronium ions.
a). If the pH of a solution is 4.2, what is the concentration of the hydronium ions?
b). A strong acid has a pH of less than 3. If the concentration of the hydronium ions is 1.6x10^-4in a particular solution, is the solution a strong acid? Explain.

5. a).Given the functions f(x) = x + 2 and g(x) = 3^x, determinean equation for (f∘ g)(x) and (g ∘ f)(x).
b).Determine f(g(3)) and g(f(3)).
c).Determine all values of x for which f(g(x)) = g(f(x)).

I don't understand these questions and need help to undertand.Please show all work. Thank you

Answer 1

Not sure what "the family" is, since there are cubics with 1 or 3 zeroes ...
I guess I'll just stick with y = x^3
#1 You know y=x^3 goes through (0,0), so lift it up by 5: y = x^3 + 5

#2 You know that y = x^3 goes through (3,27)
so, shift it down 55, to y = x^3 - 55

#3 6cos^2x +7cosx-5=0
(2cosx - 1)(3cosx + 5) = 0
cosx = 1/2 or -5/3
since |cosx| < 1, the only solution is cosx = 1/2
so your reference angle is x = π/3. cosx > 0 in QI and QIV, so
x = ±π/3 + k*2π = -5π/3, -π/3, π/3, 5π/3