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AP Calculus AB: 1.2.1 Functions

Mathematics16 CardsCreated 8 months ago

This content introduces the concept of functions as mathematical machines that map each input to exactly one output. It covers function notation, how to evaluate functions for numbers and expressions, and provides examples using equations and graphs to reinforce the concept.

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Functions

  • A function pairs one object with another. A function will produce only one object for any pairing.

  • A function can be represented by an equation. To evaluate the function for a particular value, substitute that value into the equation and solve.

  • You can evaluate a function for an expression as well as for a number. Substitute the entire expression into the equation of the function. Be careful to include parentheses where needed

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Key Terms

Term
Definition

Functions

  • A function pairs one object with another. A function will produce only one object for any pairing.

  • A function can be represe...

note

  • A function is a mathematical machine that takes one value and produces another one. In the example of an ATM machine, each account number m...

The amount of money in Brian’s savings account is given by the function M (t) = 50t^ 2 + 100t + 80, where t is the time in years. Approximately how many years will it take Brian to save $1,000?

None of the above

A function is defined as f (x) = x 2 − 5x + 3. Evaluate f (1).

f (1) = −1

A function is defined as f (x) = −2x + $6. Evaluate f ($2.20).

f ($2.20) = $1.60


A function is defined as

f (x) = 3x^ 3 − 4. Evaluate f (2).

f (2) = 20

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AP Calculus AB: 1.2.1 Functions
TermDefinition

Functions

  • A function pairs one object with another. A function will produce only one object for any pairing.

  • A function can be represented by an equation. To evaluate the function for a particular value, substitute that value into the equation and solve.

  • You can evaluate a function for an expression as well as for a number. Substitute the entire expression into the equation of the function. Be careful to include parentheses where needed

note

  • A function is a mathematical machine that takes one value and produces another one. In the example of an ATM machine, each account number matches up to exactly one balance.

  • Here the function machine is called f. f takes a value x and returns another value f(x).

  • This notation is an improvement over y-notation. It allows you to write f(5) to mean “the value of the function when x equals 5.”

  • The symbol f(5) is read as “f of 5.”

  • If you have a function whose inputs are numbers, then you can also use variables to represent those numbers.

  • For example, f(a) produces the value of the function f when the value of a is used as the input.

  • You can even evaluate a function for a number that is
    represented by an expression such as a + b. In this example, make sure to replace every appearance of x with the expression a + b. If x is squared, you must square the entire expression. If x is multiplied by 2, you must multiply the entire expression by 2. Use parentheses to help you keep track.

  • The most common name for a function is f, but sometimes it makes sense to name a function g, p, v, or even something else.

The amount of money in Brian’s savings account is given by the function M (t) = 50t^ 2 + 100t + 80, where t is the time in years. Approximately how many years will it take Brian to save $1,000?

None of the above

A function is defined as f (x) = x 2 − 5x + 3. Evaluate f (1).

f (1) = −1

A function is defined as f (x) = −2x + $6. Evaluate f ($2.20).

f ($2.20) = $1.60


A function is defined as

f (x) = 3x^ 3 − 4. Evaluate f (2).

f (2) = 20

Given the graph of f (x), find the best estimate of f (3).

−2

Given the graph of f (x), find the best estimate of f (2).

f (2) = 1

If f (x) = 3x ^2 − 10, which of the following is the new function defined by g (x) = f (x − 1)?

g (x) = 3x ^2 − 6x − 7

If h (t) = 50t ^5 + 50t ^3 + 50t, what is h (COW)?

h (COW) = 50 (COW^)5 + 50 (COW)^3 + 50 (COW)

If g (x) = −2x + 7, which of the following is the new function defined by h (x) = g (2x ^2 + 1)?

h (x) = −4x^ 2 + 5

Suppose the function

g(x) = 4x^3 - 3x - (2x-1)/5x+4

find g(-4)

g(-4) = -244 and 9/16

Given that T ( y) = y^2 − 3y + 5, compute T (x + Δ x).

T (x + Δ x) = x^ 2 + 2 xΔ x + (Δx)^ 2 − 3x − 3Δ x + 5

Rob’s height from birth to 15 years is modeled by the function h (t) = 0.24t^  2 + 22, where t is his age in years, and h (t) is his height in inches. At what age is Rob 76 inches tall?

15 years


Given the graph of g (x), find the best estimate of g (4)

1

Maria’s height from birth to 12 years is modeled by the function h (t) = 0.26t^ 2 + 22, where t is her age in years, and h (t) is her height in inches. What is Maria’s height when she is 10 years old?

None of the above