Back to Exercise: Relate domain to graph and context

Exercises: Relate the Domain of a Function to Its Graph and Quantitative Context

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Grade 9·21 problems·~30 min·Common Core Math - HS Functions·group·hsf-if-b-5

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Recall / Warm-Up

Which of the following correctly expresses "all real numbers greater than or equal to 3" in interval notation?

$( 3, \infty )$

$[ 3, \infty )$

$( -\infty, 3 ]$

$( -\infty, 3 )$

Which expression is undefined for some real number value of $x$ ?

$f(x) = 3x - 7$

$g(x) = x^2 + 4$

$h(x) = \dfrac{1}{x + 2}$

$k(x) = |x| + 1$

A graph of a function has an open circle at $( 3, 5 )$ . What does this mean?

The function is undefined for all $x$ near 3.

The point $( 3, 5 )$ is excluded — $x = 3$ is not in the domain.

The function has a vertical asymptote at $x = 3$ .

The output at $x = 3$ is zero.

Fluency Practice

What is the natural domain of $f(x) = \dfrac{5}{x - 4}$ ? Express your answer in interval notation.

$( -\infty, 4 ) \cup ( 4, \infty )$

$( -\infty, \infty )$

$( 4, \infty )$

$( -\infty, 4 ]$

What is the natural domain of $g(x) = \sqrt{x - 3}$ ? Express your answer in interval notation.

$( -\infty, \infty )$

$( -\infty, 3 )$

$[ 3, \infty )$

$( 3, \infty )$

What is the natural domain of $m(x) = \dfrac{\sqrt{x + 1}}{x - 5}$ ?

$[ -1, \infty )$

$( -\infty, \infty )$

$[ -1, 5 ) \cup ( 5, \infty )$

$( -1, 5 ) \cup ( 5, \infty )$

A graph of $f$ is shown with a solid dot at $x = -2$ and a solid dot at $x = 6$ , with a connected curve between them and no other points. What is the domain of $f$ ?

$( -2, 6 )$

$[ -2, 6 ]$

$\{ -2, 6 \}$

$( -\infty, \infty )$

A taxi charges $3 per passenger. The function $R(n) = 3n$ gives total revenue in dollars when $n$ passengers ride. The taxi holds at most 4 passengers. Which best describes how to graph $R$ on its contextual domain?

Draw a solid line through the origin with slope 3.

Draw isolated dots at $n = 0, 1, 2, 3, 4$ .

Draw a ray starting at the origin extending to the right.

Draw a solid line from $( 0, 0 )$ to $( 4, 12 )$ .

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