📝 Chapter 8: Combining Functions

Given f(x)=x²+1 and g(x)=3x-2, evaluate: a) (f+g)(4) b) (f·g)(-1) c) (f/g)(2)

Given f(x)=2x+3 and g(x)=x²-1, find: a) f(g(x)) b) g(f(x)) c) State the domain of each.

Given f(x)=√(x-1) and g(x)=x², determine the domain of f(g(x)).

If h(x) = 1/(x²+2x-3), express h as a composition of f(x)=1/x and some g(x).

Sketch y = f(x) + g(x) given tables of values for f and g at integer points from -2 to 4.

If f(x) = 2x+1 and g(x) = (x-1)/2, show that f and g are inverses by computing f(g(x)) and g(f(x)).

Decompose h(x) = sin(x²+1) into f(g(x)) in TWO different ways. Verify both decompositions give the same result.

If f(x) = x² and g(x) = |x|, is f(g(x)) = g(f(x))? Investigate and explain why or why not.

Find functions f and g such that f(g(x)) = 4x²-12x+9. Hint: recognize a pattern.

Explain the difference between f(g(x)) and g(f(x)) using a real-world analogy (e.g., getting dressed: socks then shoes vs shoes then socks). Then give a mathematical example.

Create a concept map connecting: sum of functions, difference of functions, product, quotient, and composition. For each, show: formula, domain rule, and one example.

A student says "the domain of f/g is just the intersection of the domains of f and g." What is missing from this statement? Correct it with a clear explanation.

Write a reflection (5-8 sentences): How does combining functions allow us to model more complex real-world situations than single functions alone?

A store offers a 20% discount f(x) = 0.80x, then charges 13% tax g(x) = 1.13x. a) Find (g∘f)(x) and (f∘g)(x). b) Which saves the customer money: discount then tax, or tax then discount? Show algebraically.

A tank is being filled by a pipe: V₁(t) = 50t litres, and drained by another: V₂(t) = 20t litres. a) Write the net volume as a combined function. b) If the tank starts with 200L and has capacity 1000L, when is it full? c) What if the drain only opens after t=10?

Temperature conversion: F(x) = (9/5)x + 32 converts Celsius to Fahrenheit. If outdoor temperature in Celsius is modeled by C(t) = 15 + 8sin(πt/12), find F(C(t)) — the temperature in Fahrenheit as a function of time.

A ball is thrown upward with height h(t) = -5t²+20t+2. Its shadow's position is d(t) = 3t. a) Write the distance from the ball to its shadow as a function of t. b) At what time is this distance greatest?