For the function defined by f(x)=5x-4, find a formula for f-^1(x).Replace by Interchange x and y Solve for y Replace by ------------------------------- Solve the equation 4^(2x+1) = 2^(3x+6) ? Write as Simplify the left side Now use the principle that if two powers of the same positive base are equal, and the base is not 1, then the exponents must be equal: ---------------------------------------- Find an exponential function of the form with y-intercept 2, horizontal asymptote y=-2, that passes through the point P(1,4). The graph of has horizontal asymptote y = 0, the x-axis. To move that asymptote down to y=-2 we must shift the graph downward by 2 units, so we add -2 to the right side, which makes c = -2 So far we have To have y-intercept 2, it must pass through the point (0,2) so that f(0) = 2. So far we have To pass through the point (1,4) we have f(1) = 4. So we have ------------------------------------------------- Solve the equation (f o g)^-1(3)? You didn't give f(x) and g(x), so I can't do that one without that information. ------------------------------------------------- For the function defined by , , use a sketch to help find a formula for . First we sketch the graph of , Now we draw in the identity line whose equation is Now we draw the curve reflecting it into the identity line and we get the The blue curve is the graph of To find an equation for we proceed as in the first problem above Replace by Interchange x and y Solve for y =± We can see from the graph that the blue curve is above the x-axis so we ignore the - solution Replace by ----------------------------------------- The amount of a radioactive tracer remaining after t days is given by A = A0e-0.058t, where A0 is the starting amount at the beginning of the time period. How many days will it take for one half of the original amount to decay? This is when A = A0/2 So substitute A0/2 for A A0/2 = A0e-0.058t Divide both sides by A0 1/2 = e-0.058t Use principle that X = eY can be written as Y = ln(X) to rewrite that as -0.058t = ln(1/2) t = ln(1/2)/(-0.058) = 11.95081346 days or about 12 days. ----------------------------------------------- The population P of a certain culture is expected to be given by a model P=100ert where r is a constant to be determined and t is a number of days since the original population of 100 was established. Find the value of r if the population is expected to reach 200 in 3 days. Substitute 200 for P and 3 for t P = 100ert 200 = 100er·3 200 = 100e3r Divide both sides by 100 2 = e3r Use principle that X = eY can be written as Y = ln(X) to rewrite that as 3r = ln(2) r = ln(2)/3 = .2310490602 Edwin