Mathematics 133 – Intensive Calculus for Science 1 Discussion 2 – “New Functions From Old Ones” September 13, 2005

Background

We have introduced the idea of shifting and stretching graphs to produce new graphs. Recall the idea: In y = f(x), if we replace y by y − k, then the graph is shifted up if k > 0 and down if k < 0. Similarly, if we replace x by x − h, then the graph is shifted right if h > 0 and left if h < 0. Replacing f(x) by cf(x) stretches the graph vertically if |c| > 1, shrinks it if |c| < 1, and also reflects the graph across the x axis if c < 0.

We have also discussed function composition and inverse functions. Today, we want to practice using some of these ideas.

Discussion Questions

A) Sketch the graphs y = f(x − 3) + 1 and y = −2f(x + 2) for f(x) given by the graph below. Only show the portion of the graph for the interval of x-values −2 ≤ x ≤ 2 in each case.

B) Let f(x) = −3x + 5, g(x) = \frac{1}{x^{2}+x}. What are the composite functions f(g(x)) and g(f(x))? Give the formulas and simplify.

C) Let f(x) be the temperature in degrees Celsius when the mercury in a particular thermometer is x centimeters long. Explain in practical terms the meaning of the value f(10). Why would we expect f to be invertible? Explain in practical terms the meaning of the value f^{−1}(30).

D) You are the owner of a taqueria with a very simple menu: just one type of burrito. You have fixed costs of $200 per day (rental of the shop, electricity, wages for your cashier). The ingredients for one burrito cost $4. If you sell x burritos per day, what is your total cost function C(x)? Your function should be invertible. Find a formula for the inverse function and explain in practical terms what it tells you.

Assignment

Group writeups due in class tomorrow, Wednesday, September 14.