# Modeling with Functions

Modeling with functions is a fundamental concept in mathematics and various scientific disciplines, wherein relationships between variables are represented using mathematical functions. These functions serve as powerful tools to describe, analyze, and predict real-world phenomena in fields such as physics, engineering, economics, and biology. By expressing relationships through equations, modeling with functions enables researchers, engineers, and analysts to understand complex systems, make informed decisions, and solve practical problems. Through the manipulation and interpretation of these mathematical representations, insights into the behavior and dynamics of systems can be gained, facilitating advancements in numerous domains.

- How can a function model value?
- If #f(x)=10x+3# and #g(x)=0.1x+0.3#, what is #f(g(x))#? What is #g(f(x))#? Are they inverse functions>
- If f(x) = 7-2x and g(x) = x+3 . A. What is g^-1 (x) ? B. What is f(g^-1(5)) ?
- If #(-1, 0)# lies on the graph of #y = f(x)#, what is the point that lies on the graph of #y = f(x + 3)#?
- Is #e^x# the unique function of which derivative is itself? Can you prove it?
- How to find the equation of the inverse function (the one below)?
- What is the equation for a billing process that has a one-time fee of $100 and a monthly charge of $25?
- How is growth related to the slope of a linear function?
- Which type of function should be used to model bacterial growth?
- What type of function should be used to model the interest on a bank account?
- If a cylindrical can has a surface area of 60 square inches, how do you express the volume of the can as a function of the radius?
- A bank account yields 7 percent interest, compounded annually. If you deposit $1,000 in the account, what will the account balance be after 5 years?
- The person walks 10 miles in total. If w represents the (variable) distance west she walks, and D represents her (variable) distance from home at the end of her walk, is D a function of w?
- How do you prove algebraically (not graphically): #f (x) =(2x-3)/(3x-2)# is one to one function?
- Which of the following four equations defines a function?
- Given #f(x)= 5x-4# and #g(x)=7x + 6#, what is #g(f(x))#?
- Consider the function #f(x) = x^2- 5#. If #g(x) = f(x-7)#, what can be said of #g(x)#?
- The points #(0,3),(2,-1),(4,-2)" and(-1,-1)# represent discreet or continuous data?
- The graph below shows the percentage of students enrolled in the College of Engineering at State University. Use the graph to answer the question. Does the graph represent a function? Explain
- What is f(g(x)) if f(x)=2x+5 and g(x)=x-7 and x=-3?