0. Functions

Composite functions

Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.

Find ƒ(g(h( x))).

Composite functions

Let ƒ(x) = x³, g (x) = sin x and h(x) = √x.

Find the domain of g o ƒ.

Composite functions

Let ƒ(x) = x³, g (x) = sin x and h(x) = √x .

Find the domain of ƒ o g.

{Use of Tech} Polynomial calculations

Find a polynomial ƒ that satisfies the following properties. (Hint: Determine the degree of ƒ; then substitute a polynomial of that degree and solve for its coefficients.)

ƒ(ƒ(x)) = 9x - 8

{Use of Tech} Polynomial calculations

Find a polynomial ƒ that satisfies the following properties. (Hint: Determine the degree of ƒ; then substitute a polynomial of that degree and solve for its coefficients.)

ƒ(ƒ(x)) = x⁴ - 12x² + 30

Evaluate and simplify the difference quotients (f(x + h) - f(x)) / h and (f(x) - f(a)) / (x - a) for each function.

f(x) = x² - 2x

Evaluate and simplify the difference quotients (f(x + h) - f(x)) / h and (f(x) - f(a)) / (x - a) for each function.

f(x) = 7 / (x + 3)

Simplify the difference quotient ƒ(x+h)-ƒ(x)/h

ƒ(x) = 10

Simplify the difference quotient ƒ(x+h)-ƒ(x)/h

ƒ(x) = 4x-3

Simplify the difference quotient ƒ(x+h)-ƒ(x)/h

ƒ(x) = 2x² -3x +1

Simplify the difference quotient ƒ(x+h)-ƒ(x)/h

ƒ(x) = (x)/(x+1)

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

ƒ(x) = 4 - 4x + x²

Simplify the difference quotients ƒ(x+h) - ƒ(x) / h and ƒ(x) - ƒ(a) / (x-a) by rationalizing the numerator.

ƒ(x) = - (3/√x)

Another way to approximate derivatives is to use the centered difference quotient: f' (a) ≈ f(a+h) - f(a- h) / 2h. Again, consider f(x) = √x.

c. Explain why it is not necessary to use negative values of h in the table of part (b).

Inverse of composite functions

c. Explain why if g and h are one-to-one, the inverse of ƒ(x) = g(h(x)) exists.