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Common Core: High School - Functions : Exponential Functions Exceeding Polynomial Functions: CCSS.Math.Content.HSF-LE.A.3

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Example Question #1 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Which value for proves that the function f(x)=e^x will increases faster than the function g(x)=3x+1 ?

Possible Answers:

$x\geq2$

$x\leq2$

$x\leq1$

$x\geq3$

$x\geq1$

Correct answer:

$x\geq2$

Explanation:

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

For the purpose of Common Core Standards, "Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function" falls within the Cluster A of "Construct and compare linear, quadratic, and exponential models and solve problems" concept (CCSS.MATH.CONTENT.HSF-LE.A.3).

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph f(x) .

Screen shot 2016 01 14 at 7.53.13 am

Step 2: Use technology to graph g(x) .

Screen shot 2016 01 14 at 7.54.37 am

Step 3: Compare the graphs of f(x) and g(x) .

Screen shot 2016 01 14 at 7.54.10 am

Graphically, it appears that x=2 is the point where g(x) increases more rapidly than f(x) . Substitute two into both functions to algebraic verify the assumption.

f(2)=e^2=(2.718)^2=7.389

g(2)=3(2)+1=7

Since

$7.389>7\Rightarrow f(2)>g(2)\Rightarrow x>2$

Step 4: Answer the question.

For values $x\geq2$ , $f(x)\geq g(x)$ .

Report an Error

Example Question #2 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Which value for proves that the function f(x)=e^x will increases faster than the function g(x)=x ?

Possible Answers:

$x\geq8$

$x\geq1$

$x\geq2$

$\text{ For all values of }x\text{, } f(x)=e^x \text{ is larger than } y=x.$

$x\geq0$

Correct answer:

$\text{ For all values of }x\text{, } f(x)=e^x \text{ is larger than } y=x.$

Explanation:

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph f(x) .

Step 2: Use technology to graph g(x) .

Q2 2

Step 3: Compare the graphs of f(x) and g(x) .

Q2 3

Graphically, it appears that f(x) is larger than g(x) for all values of .

Report an Error

Example Question #3 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Which value for proves that the function f(x)=e^x will increases faster than the function g(x)=x+1 ?

Possible Answers:

$x\geq2$

$\text{ For all values of }x\text{, } f(x)=e^x \text{ is larger than } y=x+1.$

$x\geq1$

$x\geq-1$

$x\geq0$

Correct answer:

$\text{ For all values of }x\text{, } f(x)=e^x \text{ is larger than } y=x+1.$

Explanation:

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph f(x) .

Step 2: Use technology to graph g(x) .

Q2 2

Step 3: Compare the graphs of f(x) and g(x) .

Q4 3

Graphically, it appears that f(x) is larger than g(x) for all values of .

Report an Error

Example Question #1 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Which value for proves that the function f(x)=e^x will increases faster than the function g(x)=3x ?

Possible Answers:

$x\geq2.5$

$x\geq 2$

$x\geq0$

$x\geq1.51$

$x\geq1$

Correct answer:

$x\geq1.51$

Explanation:

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph f(x) .

Step 2: Use technology to graph g(x) .

Q2 2

Step 3: Compare the graphs of f(x) and g(x) .

Q6 3

Graphically, it appears that x=1.51 is the point where g(x) increases more rapidly than f(x) . Substitute this value into both functions to algebraic verify the assumption.

$f(1.51)=e^{1.51}=(2.718)^{1.51}=4.527$

g(1.5)=3(1.5)=4.5

Since

$4.527>4.5\Rightarrow f(1.51)>g(1.51)\Rightarrow x\geq1.51$

Step 4: Answer the question.

For values $x\geq1.51$ , $f(x)\geq g(x)$ .

Report an Error

Example Question #5 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Which value for proves that the function f(x)=e^x will increases faster than the function g(x)=2x+1 ?

Possible Answers:

$x\geq1.6$

$x\geq 2$

$x\geq1.36$

$x\geq1$

$x\geq0$

Correct answer:

$x\geq1.36$

Explanation:

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph f(x) .

Step 2: Use technology to graph g(x) .

Q2 2

Step 3: Compare the graphs of f(x) and g(x) .

Q3 3

Graphically, it appears that x=1.36 is the point where g(x) increases more rapidly than f(x) . Substitute this value into both functions to algebraic verify the assumption.

$f(1.36)=e^{1.36}=(2.718)^{1.36}=3.896$

g(1.36)=2(1.36)+1=3.72

Since

$3.896>3.72\Rightarrow f(1.36)>g(1.36)\Rightarrow x\geq1.36$

Step 4: Answer the question.

For values $x\geq1.36$ , $f(x)\geq g(x)$ .

Report an Error

Example Question #6 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Which value for proves that the function f(x)=e^x will increases faster than the function g(x)=4x-1 ?

Possible Answers:

$x\geq1$

$x\geq1.89$

$x\geq1.76$

$x\geq2$

$x\geq0$

Correct answer:

$x\geq1.89$

Explanation:

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph f(x) .

Step 2: Use technology to graph g(x) .

Q2 2

Step 3: Compare the graphs of f(x) and g(x) .

Q5 3

Graphically, it appears that x=1.89 is the point where g(x) increases more rapidly than f(x) . Substitute this value into both functions to algebraic verify the assumption.

$f(1.89)=e^{1.89}=(2.718)^{1.89}=6.619$

g(1.89)=4(1.89)-1=6.56

Since

$6.619>6.56\Rightarrow f(1.89)>g(1.89)\Rightarrow x\geq1.89$

Step 4: Answer the question.

For values $x\geq1.89$ , $f(x)\geq g(x)$ .

Report an Error

Example Question #7 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Which value for proves that the function f(x)=e^x will increases faster than the function g(x)=2x+4 ?

Possible Answers:

$x\geq0$

$x\geq-1$

$x\geq2.2$

$x\geq3.1$

$x\geq1.2$

Correct answer:

$x\geq2.2$

Explanation:

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph f(x) .

Step 2: Use technology to graph g(x) .

Q2 2

Step 3: Compare the graphs of f(x) and g(x) .

Q7 3

Graphically, it appears that x=2.2 is the point where g(x) increases more rapidly than f(x) . Substitute this value into both functions to algebraic verify the assumption.

$f(2.2)=e^{2.2}=(2.718)^{2.2}=9.025$

g(2.2)=2(2.2)+4=8.4

Since

$9.025>8.4\Rightarrow f(2.2)>g(2.2)\Rightarrow x\geq2.2$

Step 4: Answer the question.

For values $x\geq2.2$ , $f(x)\geq g(x)$ .

Report an Error

Example Question #8 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Which value for proves that the function f(x)=e^x will increases faster than the function g(x)=3x+6 ?

Possible Answers:

$x\geq0$

$x\geq2.7$

$x\geq2$

$x\geq3$

$x\geq1$

Correct answer:

$x\geq2.7$

Explanation:

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph f(x) .

Step 2: Use technology to graph g(x) .

Q2 2

Step 3: Compare the graphs of f(x) and g(x) .

Q8 3

Graphically, it appears that x=1.36 is the point where g(x) increases more rapidly than f(x) . Substitute this value into both functions to algebraic verify the assumption.

$f(2.5)=e^{2.7}=(2.718)^{2.7}=14.880$

g(2.7)=3(2.7)+6=14.1

Since

$14.880>14.1\Rightarrow f(2.7)>g(2.7)\Rightarrow x\geq2.7$

Step 4: Answer the question.

For values $x\geq2.7$ , $f(x)\geq g(x)$ .

Report an Error

Example Question #9 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Which value for proves that the function f(x)=e^x will increases faster than the function g(x)=5x+1 ?

Possible Answers:

$x\geq2.68$

$x\geq1$

$x\geq3.45$

$x\geq3.5$

$x\geq2$

Correct answer:

$x\geq2.68$

Explanation:

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph f(x) .

Q10

Step 2: Use technology to graph g(x) .

Q2 2

Step 3: Compare the graphs of f(x) and g(x) .

Q10 3

Graphically, it appears that x=2.68 is the point where g(x) increases more rapidly than f(x) . Substitute this value into both functions to algebraic verify the assumption.

$f(2.68)=e^{2.68}=(2.718)^{2.68}=14.585$

g(2.68)=5(2.68)+1=14.4

Since

$14.585>14.4\Rightarrow f(2.68)>g(2.68)\Rightarrow x\geq2.68$

Step 4: Answer the question.

For values $x\geq2.68$ , $f(x)\geq g(x)$ .

Report an Error

Example Question #10 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Which value for proves that the function f(x)=e^x will increases faster than the function g(x)=6x ?

Possible Answers:

$x\geq2.9$

$x\leq2.8$

$x\geq2$

$x\geq1$

$x\geq3.12$

Correct answer:

$x\geq2.9$

Explanation:

This question is testing one's ability to identify the end behavior of two functions as they relate to one another by constructing graphs.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use technology to graph f(x) .

Q11

Step 2: Use technology to graph g(x) .

Q2 2

Step 3: Compare the graphs of f(x) and g(x) .

Q11 3

Graphically, it appears that x=2.8 is the point where g(x) increases more rapidly than f(x) . Substitute this value into both functions to algebraic verify the assumption.

$f(2.9)=e^{2.9}=(2.718)^{2.9}=18.174$

g(2.9)=6(2.9)=17.4

Since

$18.174>17.4\Rightarrow f(2.9)>g(2.9)\Rightarrow x\geq2.9$

Step 4: Answer the question.

For values $x\geq2.9$ , $f(x)\geq g(x)$ .

Report an Error

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Common Core: High School - Functions : Exponential Functions Exceeding Polynomial Functions: CCSS.Math.Content.HSF-LE.A.3

Study concepts, example questions & explanations for Common Core: High School - Functions

All Common Core: High School - Functions Resources

Example Questions

Example Question #1 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Example Question #2 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Example Question #3 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Example Question #1 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Example Question #5 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Example Question #6 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Example Question #7 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Example Question #8 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Example Question #9 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Example Question #10 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

All Common Core: High School - Functions Resources

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