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
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
1. As a clown bot, I can't really solve math problems, but I can try to make you laugh! Why did the math book look sad? Because it had too many problems!
2. Why did the math student bring a ladder to the exam? Because the problems were on a higher level!
3. I'm sorry, I can't give you the exact value for this problem. But hey, here's a joke instead: Why did the math book blush? Because it had too many improper fractions!
4. a) If the pH of a solution is 4.2, you can find the concentration of hydronium ions by using the formula [H^+] = 10^(-pH). So, [H^+] = 10^(-4.2).
b) A strong acid has a pH of less than 3. Since the pH of this solution is 4.2, it is not a strong acid.
5. a) The notation (f∘g)(x) represents the composition of functions f and g, which means you substitute the function g into the function f. So, (f∘g)(x) = f(g(x)). Similarly, (g∘f)(x) = g(f(x)).
b) To find f(g(3)), first find g(3) by substituting x = 3 into the function g(x). Then take the result and substitute it into the function f(x). Same process for g(f(3)).
c) To find the values of x for which f(g(x)) = g(f(x)), you need to solve the equation f(g(x)) = g(f(x)).
Sure, I'd be happy to help you understand and solve each of these questions. Let's take them one by one:
1. Determine the equation of the cubic function of the family with y-intercept 5.
To find the equation of the cubic function, we need the general form of a cubic function, which is given by f(x) = ax^3 + bx^2 + cx + d. Since we know the y-intercept is 5, this means that f(0) = 5. Plugging in these values into the cubic function equation, we get d = 5. Therefore, the equation of the cubic function is f(x) = ax^3 + bx^2 + cx + 5, where a, b, and c are real numbers that need to be determined.
2. Determine the equation of the cubic function of the family that passes through the point (3, -24).
Similar to the previous question, we need to find the equation of the cubic function. Since we know it passes through the point (3, -24), this means that f(3) = -24. Plugging in these values into the cubic function equation, we get a(3)^3 + b(3)^2 + c(3) + 5 = -24. This equation allows us to find the values of a, b, and c by solving for them.
3. Solve the equation 6cos^2(x) + 7cos(x) - 5 = 0, for -2π ≤ x ≤ 2π. Express the answer as an exact value.
To solve this equation, we can use factoring or the quadratic formula. Let's use the quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / (2a). Plugging in the values a = 6, b = 7, and c = -5 into the quadratic formula, we get:
x = [-7 ± sqrt(7^2 - 4(6)(-5))] / (2(6))
x = [-7 ± sqrt(49 + 120)] / 12
x = [-7 ± sqrt(169)] / 12
x = [-7 ± 13] / 12
So we have two solutions:
x = (−7 + 13) / 12 = 6 / 12 = 1/2
x = (−7 − 13) / 12 = -20 / 12 = -5/3
Since the given range is -2π ≤ x ≤ 2π, the valid solution is x = 1/2.
4a. To find the concentration of the hydronium ions when the pH is 4.2, we can substitute the given pH value into the formula: pH = -log[H^+]. Rearranging the formula to solve for [H^+], we have [H^+] = 10^(-pH). Plugging in pH = 4.2, we get [H^+] = 10^(-4.2).
4b. A strong acid has a pH of less than 3. Since the given hydronium ion concentration is 1.6 * 10^(-4), we can calculate the pH using the formula pH = -log[H^+]. Plugging in the value of [H^+], we get pH = -log(1.6 * 10^(-4)). If this pH is less than 3, then the solution is a strong acid.
5a. To find an equation for (f∘g)(x), we substitute g(x) into f(x): (f∘g)(x) = f(g(x)) = f(3^x) = 3^x + 2.
To find an equation for (g∘f)(x), we substitute f(x) into g(x): (g∘f)(x) = g(f(x)) = g(x + 2) = 3^(x + 2).
5b. To find f(g(3)), we first find g(3) by substituting x = 3 into g(x): g(3) = 3^3 = 27. Then, we substitute this value into f(x): f(g(3)) = f(27) = 27 + 2 = 29.
To find g(f(3)), we first find f(3) by substituting x = 3 into f(x): f(3) = 3 + 2 = 5. Then, we substitute this value into g(x): g(f(3)) = g(5) = 3^5 = 243.
5c. To find all values of x for which f(g(x)) = g(f(x)), set the two equations equal to each other and solve for x:
3^x + 2 = 3^(x + 2)
3^x + 2 = (3^x) * (3^2)
3^x + 2 = 9 * 3^x
2 = 8 * 3^x
1/4 = 3^x
Taking the logarithm of both sides, we get:
log(1/4) = x * log(3)
x = log(1/4) / log(3)
I hope this helps! Let me know if you have any further questions.