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Algebra Help: Understanding Functions Domain And Range

Review real example questions for Understanding Functions Domain And Range in Algebra.

Question 1 / 10

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Let $g(x)=\dfrac{1}{x-5}$ . To keep $g$ a function with one output for each input, $g(x)$ must be defined for the input. For what values of $x$ is $g(x)$ defined? (Answer in words or interval notation.)

All questions

Question 1

All real numbers except $x=5$ (correct answer)
All real numbers except $x=0$
$[5,\infty)$
All real numbers

Explanation: This question tests your understanding of what functions are, and how to determine their domains (possible inputs) and ranges (possible outputs). The domain is the set of all possible input values (x-values) that make sense for the function: for formulas, we exclude values that would cause division by zero or square roots of negatives, and in real-world contexts, we only include values that are realistic (like you can't have -3 people or 2.5 items if they're discrete). For g(x) = 1/(x-5), we need to avoid division by zero, so the denominator x-5 cannot equal 0, which means x cannot equal 5. Choice A is correct because it states 'all real numbers except x=5', which excludes only the problematic value while allowing all others. Choice B incorrectly excludes x=0, but plugging in x=0 gives g(0)=1/(-5)=-1/5, which is perfectly defined. For finding domain from a formula: (1) Start by assuming all real numbers are okay, (2) Then look for restrictions—is there a square root (need inside ≥ 0)? A fraction (need denominator ≠ 0)? (3) Write the domain excluding or including only the values that work. Most Algebra 1 functions have domains like 'all reals' or 'x ≥ some number' or 'all reals except one value.'

Question 2

A function $f$ is defined by $f(x)=\sqrt{x-2}$ . Because a function assigns exactly one output to each input in its domain, we must restrict inputs so the output is real. What is the domain of $f$ ? (Answer in interval notation.)

$(-\infty,2]$
$[2,\infty)$ (correct answer)
$(2,\infty)$
All real numbers

Explanation: This question tests your understanding of what functions are, and how to determine their domains (possible inputs) and ranges (possible outputs). The domain is the set of all possible input values (x-values) that make sense for the function: for formulas, we exclude values that would cause division by zero or square roots of negatives, and in real-world contexts, we only include values that are realistic (like you can't have -3 people or 2.5 items if they're discrete). For f(x) = √(x-2), we need the expression under the square root to be non-negative, so x-2 ≥ 0, which means x ≥ 2. Choice B is correct because [2,∞) represents all real numbers greater than or equal to 2, using a square bracket at 2 to show it's included. Choice A would give negative values under the square root, making the output imaginary rather than real. For finding domain from a formula: (1) Start by assuming all real numbers are okay, (2) Then look for restrictions—is there a square root (need inside ≥ 0)? A fraction (need denominator ≠ 0)? (3) Write the domain excluding or including only the values that work. Most Algebra 1 functions have domains like 'all reals' or 'x ≥ some number' or 'all reals except one value.'

Question 3

The function $g$ is defined by $g(x)=\dfrac{5}{x-2}$ . For what values of $x$ is $g(x)$ defined? (Give your answer in words or set description.)

All real numbers $x\ge 2$
All real numbers except $x=5$
All real numbers except $x=2$ (correct answer)
All real numbers $x\le 2$

Explanation: This question tests your understanding of what functions are, and how to determine their domains (possible inputs) and ranges (possible outputs). The domain is the set of all possible input values (x-values) that make sense for the function: for formulas, we exclude values that would cause division by zero or square roots of negatives, and in real-world contexts, we only include values that are realistic (like you can't have -3 people or 2.5 items if they're discrete). For g(x) = 5/(x-2), we need to avoid division by zero, which happens when the denominator x - 2 = 0, so x = 2 must be excluded from the domain. Choice A is correct because it states that g(x) is defined for all real numbers except x = 2, which is exactly where the denominator becomes zero. Choice B incorrectly excludes x = 5, but plugging in x = 5 gives g(5) = 5/3, which is perfectly valid. For finding domain from a formula: (1) Start by assuming all real numbers are okay, (2) Then look for restrictions—is there a square root (need inside ≥ 0)? A fraction (need denominator ≠ 0)? (3) Write the domain excluding or including only the values that work. When you see a fraction, always check where the denominator equals zero—those are your excluded values!

Question 4

The function $p(x)=\sqrt{x}$ assigns one output to each input in its domain. What is the range of $p$ ? (Answer in interval notation.)

$(-\infty,\infty)$
$[0,\infty)$ (correct answer)
$(-\infty,0]$
$(0,\infty)$

Explanation: This question tests your understanding of what functions are, and how to determine their domains (possible inputs) and ranges (possible outputs). The range is the set of all possible output values (y-values) the function can produce: for a quadratic that opens upward with vertex at (2, -3), the range is all y-values greater than or equal to -3 because the parabola goes up from that minimum point forever. For p(x) = √x, the square root function only produces non-negative outputs—you can never get a negative number from a principal square root. The smallest output is √0 = 0, and as x increases, √x increases without bound. Choice B is correct because [0,∞) represents all non-negative real numbers, with the square bracket showing 0 is included as the minimum possible output. Choice C incorrectly suggests negative outputs are possible, but √x is always non-negative by definition. For finding range: linear functions usually have range = all real numbers. For quadratics, find the vertex first—if it opens up, range is y ≥ (vertex y-value); if it opens down, range is y ≤ (vertex y-value). For square roots, range is usually y ≥ 0. The function type tells you a lot!

Question 5

What is the domain of the function $f(x)=\sqrt{x-5}$ ? Give your answer in interval notation.

$(5,\infty)$
$(-\infty,5]$
$[5,\infty)$ (correct answer)
$(-\infty,\infty)$

Explanation: This question tests your understanding of what functions are, and how to determine their domains (possible inputs) and ranges (possible outputs). The domain is the set of all possible input values (x-values) that make sense for the function: for formulas, we exclude values that would cause division by zero or square roots of negatives, and in real-world contexts, we only include values that are realistic (like you can't have -3 people or 2.5 items if they're discrete). To find the domain of f(x) = √(x-5), set the inside of the square root greater than or equal to zero: x - 5 ≥ 0, so x ≥ 5. Choice C is correct because [5, ∞) includes x = 5 (where the output is 0) and all larger values, ensuring the square root is defined for real numbers. A common mistake is thinking it's (-∞, 5], but that would make x - 5 negative, which isn't allowed for real square roots—keep practicing identifying those restrictions! For finding domain from a formula: (1) Start by assuming all real numbers are okay, (2) Then look for restrictions—is there a square root (need inside ≥ 0)? A fraction (need denominator ≠ 0)? (3) Write the domain excluding or including only the values that work. Most Algebra 1 functions have domains like 'all reals' or 'x ≥ some number' or 'all reals except one value.'

Question 6

The function $k$ is defined by $k(x)=\sqrt{9-x}$ . What is the domain of $k$ ? (Give your answer in interval notation.)

$( -\infty,\infty)$
$[9,\infty)$
$( -\infty,9)$
$( -\infty,9]$ (correct answer)

Explanation: This question tests your understanding of what functions are, and how to determine their domains (possible inputs) and ranges (possible outputs). The domain is the set of all possible input values (x-values) that make sense for the function: for formulas, we exclude values that would cause division by zero or square roots of negatives, and in real-world contexts, we only include values that are realistic (like you can't have -3 people or 2.5 items if they're discrete). For k(x) = √(9-x), we need 9 - x ≥ 0, which means 9 ≥ x, or equivalently x ≤ 9. Choice A is correct because (-∞,9] represents all real numbers less than or equal to 9, with the square bracket showing 9 is included since √0 = 0 is valid. Choice B would be the domain if we had √(x-9) instead, requiring x ≥ 9. For finding domain from a formula: (1) Start by assuming all real numbers are okay, (2) Then look for restrictions—is there a square root (need inside ≥ 0)? A fraction (need denominator ≠ 0)? (3) Write the domain excluding or including only the values that work. When the variable is subtracted inside a square root, flip your thinking!

Question 7

The function $p$ is defined by $p(x)=-(x-1)^2+4$ . What is the range of $p$ ? (Give your answer in interval notation.)

$[4,\infty)$
$(-\infty,4]$ (correct answer)
$(-\infty,4)$
$(-\infty,\infty)$

Explanation: This question tests your understanding of what functions are, and how to determine their domains (possible inputs) and ranges (possible outputs). The range is the set of all possible output values (y-values) the function can produce: for a quadratic that opens upward with vertex at $(2, -3)$ , the range is all y-values greater than or equal to $-3$ because the parabola goes up from that minimum point forever. For $p(x) = -(x-1)^2 + 4$ , the negative sign in front means this parabola opens downward, and the vertex form shows the vertex is at $(1, 4)$ , giving a maximum value of 4. Choice B is correct because $(-\infty,4]$ represents all y-values less than or equal to 4, with the square bracket showing 4 is included since $p(1) = 4$ . Choice A would suggest the range starts at 4 and goes up, but this parabola opens down from its maximum. For finding range: linear functions usually have range = all real numbers. For quadratics, find the vertex first—if it opens up, range is $y \geq$ (vertex y-value); if it opens down, range is $y \leq$ (vertex y-value). The negative coefficient flips everything!

Question 8

The function $h$ is defined by $h(x)=x^2-9$ . What is the range of $h$ ? (Give your answer in interval notation.)

$( -\infty,\infty)$
$[-9,\infty)$ (correct answer)
$( -\infty,-9]$
$(-9,\infty)$

Explanation: This question tests your understanding of what functions are, and how to determine their domains (possible inputs) and ranges (possible outputs). The range is the set of all possible output values (y-values) the function can produce: for a quadratic that opens upward with vertex at (2, -3), the range is all y-values greater than or equal to -3 because the parabola goes up from that minimum point forever. For h(x) = x² - 9, this is a parabola that opens upward (positive coefficient on x²) with vertex at (0, -9), found by completing the square or noting that x² has its minimum at x = 0. Choice C is correct because [-9,∞) represents all y-values greater than or equal to -9, with the square bracket showing -9 is included since h(0) = -9. Choice B uses a parenthesis at -9, incorrectly suggesting -9 isn't in the range, but it is the minimum value. For finding range: linear functions usually have range = all real numbers. For quadratics, find the vertex first—if it opens up, range is y ≥ (vertex y-value); if it opens down, range is y ≤ (vertex y-value). The function type tells you a lot!

Question 9

What are the possible output values (the range) of the function $p(x)=\sqrt{x}$ ? Give your answer in interval notation.

$( -\infty, \infty)$
$( -\infty, 0]$
$[0, \infty)$ (correct answer)
$(0, \infty)$

Explanation: This question tests your understanding of what functions are, and how to determine their domains (possible inputs) and ranges (possible outputs). The range is the set of all possible output values (y-values) the function can produce: for a quadratic that opens upward with vertex at (2, -3), the range is all y-values greater than or equal to -3 because the parabola goes up from that minimum point forever. For p(x) = √x, inputs are x ≥ 0, and outputs start at 0 (when x=0) and go to infinity as x increases, never negative. Choice C is correct because [0, ∞) includes 0 and all positive reals, matching the non-negative outputs of square roots. Choice B suggests negative outputs, but square roots can't produce negatives in real numbers—think about how √x is always ≥ 0! For finding range: linear functions usually have range = all real numbers. For quadratics, find the vertex first—if it opens up, range is y ≥ (vertex y-value); if it opens down, range is y ≤ (vertex y-value). For square roots, range is usually y ≥ 0. The function type tells you a lot!

Question 10

For what values of $x$ is the function $g(x)=\dfrac{3}{x-2}$ defined? Give your answer in interval notation.

$(-\infty,2]\cup[2,\infty)$
$(-\infty,2)\cup(2,\infty)$ (correct answer)
$(-\infty,\infty)$
$[2,\infty)$

Explanation: This question tests your understanding of what functions are, and how to determine their domains (possible inputs) and ranges (possible outputs). The domain is the set of all possible input values (x-values) that make sense for the function: for formulas, we exclude values that would cause division by zero or square roots of negatives, and in real-world contexts, we only include values that are realistic (like you can't have -3 people or 2.5 items if they're discrete). For g(x) = 3/(x-2), the function is undefined when the denominator is zero, so x - 2 = 0 means x=2 is excluded, but all other real numbers work. Choice A is correct because (-∞, 2) ∪ (2, ∞) includes everything except x=2, where the function would involve division by zero. If you chose B thinking it's all reals, that's understandable, but remember to always check for denominator zeros—it's a key restriction! For finding domain from a formula: (1) Start by assuming all real numbers are okay, (2) Then look for restrictions—is there a square root (need inside ≥ 0)? A fraction (need denominator ≠ 0)? (3) Write the domain excluding or including only the values that work. Most Algebra 1 functions have domains like 'all reals' or 'x ≥ some number' or 'all reals except one value.'