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Algebra Help: Function Notation And Evaluation

Review real example questions for Function Notation And Evaluation in Algebra.

Question 1 / 10

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If $m(x)=2x^2-3x+1$ , evaluate $m(-1)$ .

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Question 1

If $m(x)=2x^2-3x+1$ , evaluate $m(-1)$ .

$0$
$2$
$4$
$6$ (correct answer)

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. To evaluate a function like m(x) = 2x² - 3x + 1 at a specific value, we replace every x with that value and calculate: m(-1) means substitute -1 for x, giving 2(-1)² - 3(-1) + 1. When evaluating with a negative number like m(-1), we need to be extra careful with signs! Substituting -1 for x in m(x) = 2x² - 3x + 1, we get m(-1) = 2(-1)² - 3(-1) + 1, which equals 2(1) + 3 + 1 = 2 + 3 + 1 = 6. Notice how the parentheses around -1 help keep track of the negative! Choice D is correct because it properly substitutes -1 for x in the function and calculates accurately: 2(-1)² - 3(-1) + 1 = 2(1) + 3 + 1 = 6. Nice work if you got this! Choice A (0) might result from sign errors, possibly calculating 2(-1) - 3(-1) + 1 without squaring the first term. Remember to follow the function formula exactly as written! When substituting negative numbers, always use parentheses to protect yourself from sign errors: write m(-1) = 2(-1)² - 3(-1) + 1. The parentheses keep everything clear! Remember that (-1)² = 1 and -3(-1) = +3, so negative inputs can lead to positive terms!

Question 2

The cost in dollars to rent a bike for $x$ hours is $C(x)=8x+12$ . What is $C(3)$ ?

$36$ (correct answer)
$20$
$8$
$24$

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. In real-world contexts, a function like C(x) = 8x + 12 gives us a formula to calculate one quantity (like cost) from another (like number of hours), and evaluating C(3) tells us the specific cost when 3 hours are involved. Starting with C(x) = 8x + 12 and finding C(3), we substitute 3 for x everywhere: C(3) = 8(3) + 12. Now we calculate step by step: 8(3) + 12 = 24 + 12 = 36. In this problem, C(3) = 36 dollars, which means the cost to rent a bike for 3 hours is $36. The function helps us quickly answer 'what if' questions by just plugging in different input values! Choice B is correct because it properly substitutes 3 for x in the function and calculates accurately: 8(3) + 12 = 36. Nice work if you got this! Choice C has the right idea but makes an arithmetic error, calculating 8(3) = 24 but then forgetting to add the 12. Double-checking your arithmetic is always a good idea! In word problems, always state what your answer means: don't just write 'C(3) = 36'—say 'C(3) = 36 dollars, which is the total cost to rent a bike for 3 hours.' This shows you understand what the math represents!

Question 3

For $f(x)=4-2x$ , what is the value of $f(0)$ ?

$-2$
$0$
$2$
$4$ (correct answer)

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. To evaluate a function like f(x) = 4 - 2x at a specific value, we replace every x with that value and calculate: f(0) means substitute 0 for x, giving 4 - 2(0). Starting with f(x) = 4 - 2x and finding f(0), we substitute 0 for x everywhere: f(0) = 4 - 2(0). Now we calculate step by step: 2(0) = 0, then 4 - 0 = 4. Choice D is correct because it properly substitutes 0 for x in the function and calculates accurately: 4 - 2(0) = 4 - 0 = 4. Nice work if you got this! Choice B (0) treats f(0) as multiplication (f times 0), but function notation uses parentheses to show the input, not to multiply. Think of f as a 'machine' that processes inputs, not as a number to multiply with. Here's the foolproof way to evaluate functions: (1) write out the function formula, (2) wherever you see the variable, write the input value in parentheses, (3) calculate step by step using order of operations. When the input is 0, any term with x becomes 0, making the calculation especially simple!

Question 4

A parking garage charges based on the function $P(t) = 5 + 2⌊t⌋$ where $t$ is time in hours and $⌊t⌋$ is the greatest integer less than or equal to $t$ .

Sarah parks for 2.7 hours. If the equation $P(2.7) = c$ represents her total cost, what does the expression $P(2.7) - P(1.3)$ represent?

The difference in parking time between Sarah's visit and a 1.3-hour visit
The total cost for parking 1.4 hours at the garage's standard rate
The average hourly rate Sarah pays for the extra 1.4 hours of parking
The additional cost Sarah pays compared to someone who parks for 1.3 hours (correct answer)

Explanation: When you encounter function notation with operations between two function values, focus on what each piece represents and how their combination creates meaning in the real-world context. Let's calculate each value using $P(t) = 5 + 2⌊t⌋$ . For Sarah's 2.7-hour visit: $P(2.7) = 5 + 2⌊2.7⌋ = 5 + 2(2) = 9$ dollars. For a 1.3-hour visit: $P(1.3) = 5 + 2⌊1.3⌋ = 5 + 2(1) = 7$ dollars. Therefore, $P(2.7) - P(1.3) = 9 - 7 = 2$ dollars. This difference represents how much more Sarah pays compared to someone parking for 1.3 hours, making D correct. Choice A confuses the mathematical operation with what's being measured. The expression $P(2.7) - P(1.3)$ involves costs, not time differences. While the time difference is indeed $2.7 - 1.3 = 1.4$ hours, that's not what this expression calculates. Choice B misinterprets the result. The $2 difference isn't the cost for parking 1.4 hours total—it's the additional cost between two different parking durations. Choice C incorrectly suggests we're finding an average rate. An average would require dividing by time, but this expression only subtracts costs. Study tip: When you see function differences like $ $f(a) - f(b)$ $, always ask "What does this tell me about the relationship between the two scenarios?" The difference between function outputs typically represents how much one situation differs from another in terms of the function's measured quantity.

Question 5

A delivery service charges according to the function $C(w) = 12 + 3w$ where $w$ is the weight in pounds and $C(w)$ is the cost in dollars.

If a customer pays $27 for shipping, and the statement "$ $C(w) = 27$ $" represents this situation, what does the expression "$ $C(w + 2)$ $" represent in this context?

The cost to ship a package that weighs 2 pounds more than the original package (correct answer)
The cost to ship the original package plus an additional $2 service fee
The total cost for shipping two packages of the same weight as the original
The cost per pound for a package that weighs 2 pounds more than the original

Explanation: $C(w + 2)$ means we substitute $(w + 2)$ for $w$ in the original function, giving $C(w + 2) = 12 + 3(w + 2)$ . This represents the cost for a package that weighs 2 pounds more than the weight $w$ . Choice B incorrectly interprets the $+2$ as an additional fee rather than additional weight. Choice C misunderstands this as shipping multiple packages. Choice D incorrectly suggests this gives a per-pound rate rather than a total cost.

Question 6

For a function $p(x)$ , it is known that $p(5) = 8$ , $p(8) = 2$ , and $p(2) = 5$ . What is the value of $p(p(p(5)))$ ?

$2$
$5$ (correct answer)
$8$
$15$

Explanation: We evaluate from the inside out. First, $p(5) = 8$ . Then $p(p(5)) = p(8) = 2$ . Finally, $p(p(p(5))) = p(2) = 5$ . Choice A ( $2$ ) represents stopping at $p(p(5))$ , Choice C ( $8$ ) represents stopping at $p(5)$ , and Choice D ( $15$ ) represents incorrectly adding the intermediate values.

Question 7

Let $f(x) = 2x - 1$ and $g(x) = x^2 + 3$ . If $f(a) = 9$ and $g(b) = 12$ , what is the value of $f(b) + g(a)$ ?

$27$
$42$
$36$
$33$ (correct answer)

Explanation: This problem tests function evaluation and substitution—core skills where you plug values into functions and work with the results strategically. Start by finding the values of $a$ and $b$ . Since $f(a) = 9$ , substitute into $f(x) = 2x - 1$ : $2a - 1 = 9$ , so $2a = 10$ and $a = 5$ . Since $g(b) = 12$ , substitute into $g(x) = x^2 + 3$ : $b^2 + 3 = 12$ , so $b^2 = 9$ and $b = 3$ (taking the positive solution). Now calculate $f(b) + g(a)$ . With $b = 3$ : $f(3) = 2(3) - 1 = 5$ . With $a = 5$ : $g(5) = 5^2 + 3 = 28$ . Therefore, $f(b) + g(a) = 5 + 28 = 33$ . Choice A ( $27$ ) likely comes from incorrectly calculating $g(5) = 25 - 1 + 3 = 27$ , mixing up the function formulas. Choice B ( $42$ ) might result from computing $f(5) + g(3) = 9 + 12 = 21$ and doubling it, or from other calculation errors. Choice C ( $36$ ) could come from finding $a = 5$ and $b = 3$ correctly but then computing $ab + f(b) = 15 + 5 + 16 = 36$ , confusing what expression to evaluate. The key strategy: solve for the unknown inputs first, then carefully substitute into the requested expression. Always double-check which function you're applying to which variable—it's easy to mix up $f(a)$ versus $f(b)$ when working quickly.

Question 8

A function $m(x)$ satisfies $m(x + 3) = 2m(x) + 1$ for all values in its domain. If $m(1) = 4$ , what is $m(7)$ ?

$9$
$17$
$19$ (correct answer)
$35$

Explanation: We use the recursive relationship $m(x + 3) = 2m(x) + 1$ . Starting with $m(1) = 4$ : $m(4) = m(1 + 3) = 2m(1) + 1 = 2(4) + 1 = 9$ . Then $m(7) = m(4 + 3) = 2m(4) + 1 = 2(9) + 1 = 19$ . Choice A represents $m(4)$ only, Choice B represents forgetting to add 1 in the final step, and Choice D represents incorrectly multiplying by 3 instead of applying the relationship twice.

Question 9

Given $h(x) = x^2 + 1$ , what is $h(x + 2)$ ?

$x^2 + 3$
$(x+2)^2 + 1$ (correct answer)
$x^2 + 4x + 1$
$x^2 + 2x + 3$

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. Function notation h(x) tells us the rule for calculating outputs from inputs: when you see h(x + 2), it means 'substitute (x + 2) for every x in the function formula,' like filling in a blank everywhere you see x. For h(x + 2), we substitute the entire expression (x + 2) for every x: h(x + 2) = (x + 2)² + 1. This gives us (x + 2)² + 1, which is the expression as is. Choice B is correct because it properly substitutes (x + 2) for x in the function: (x + 2)² + 1. Nice work if you got this! Choice C is a common slip-up: it expands the expression to x² + 4x + 4 + 1 = x² + 4x + 5 but the question asks for h(x + 2) without specifying to expand, though B keeps it in the substituted form. Remember, sometimes the answer is the substituted expression itself! To check your work, ask yourself: 'Did I replace every single x with my input value?' and 'Did I follow PEMDAS for my calculation?' These two checks catch most mistakes!

Question 10

Given $g(x) = 3x^2 + 2x - 1$ , what is $g(-1)$ ?

$0$ (correct answer)
$-6$
$4$
$2$

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. To evaluate a function like $g(x) = 3x^2 + 2x - 1$ at a specific value, we replace every x with that value and calculate: $g(-1)$ means substitute -1 for x, giving $3(-1)^2 + 2(-1) - 1 = 3(1) - 2 - 1 = 0$ . When evaluating with a negative number like $g(-1)$ , we need to be extra careful with signs! Substituting -1 for x in $g(x) = 3x^2 + 2x - 1$ , we get $3(-1)^2 + 2(-1) - 1$ , which equals $3(1) - 2 - 1 = 0$ . Notice how the parentheses around -1 help keep track of the negative! Choice A is correct because it properly substitutes -1 for x in the function and calculates accurately: $3 - 2 - 1 = 0$ . Nice work if you got this! Choice C is a common slip-up: it makes a sign error, perhaps calculating $3(-1)^2$ as $3(-1) = -3$ or mishandling the signs. When you substitute a negative number, use parentheses to keep track: $(-1)^2$ is positive, but forgetting can lead to errors. When substituting negative numbers, always use parentheses to protect yourself from sign errors: write $f(-3) = 2(-3)^2 + 1$ , not $f(-3) = 2-3^2 + 1$ . The parentheses keep everything clear!