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

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Question

Which value for proves that the function will increases faster than the function ?

Answer

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 .

Step 2: Use technology to graph .

Step 3: Compare the graphs of and .

Graphically, it appears that is the point where increases more rapidly than . Substitute two into both functions to algebraic verify the assumption.

Since

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

Step 4: Answer the question.

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

Compare your answer with the correct one above

Question

Which value for proves that the function will increases faster than the function ?

Answer

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 .

Step 2: Use technology to graph .

Step 3: Compare the graphs of and .

Graphically, it appears that is larger than for all values of .

Compare your answer with the correct one above

Question

Which value for proves that the function will increases faster than the function ?

Answer

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 .

Step 2: Use technology to graph .

Step 3: Compare the graphs of and .

Graphically, it appears that is the point where increases more rapidly than . Substitute this value into both functions to algebraic verify the assumption.

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

Since

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

Step 4: Answer the question.

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

Compare your answer with the correct one above

Question

Which value for proves that the function will increases faster than the function ?

Answer

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 .

Step 2: Use technology to graph .

Step 3: Compare the graphs of and .

Graphically, it appears that is larger than for all values of .

Compare your answer with the correct one above

Question

Which value for proves that the function will increases faster than the function ?

Answer

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 .

Step 2: Use technology to graph .

Step 3: Compare the graphs of and .

Graphically, it appears that is the point where increases more rapidly than . Substitute this value into both functions to algebraic verify the assumption.

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

Since

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

Step 4: Answer the question.

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

Compare your answer with the correct one above

Question

Which value for proves that the function will increases faster than the function ?

Answer

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 .

Step 2: Use technology to graph .

Step 3: Compare the graphs of and .

Graphically, it appears that is the point where increases more rapidly than . Substitute this value into both functions to algebraic verify the assumption.

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

Since

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

Step 4: Answer the question.

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

Compare your answer with the correct one above

Question

Which value for proves that the function will increases faster than the function ?

Answer

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 .

Step 2: Use technology to graph .

Step 3: Compare the graphs of and .

Graphically, it appears that is the point where increases more rapidly than . Substitute this value into both functions to algebraic verify the assumption.

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

Since

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

Step 4: Answer the question.

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

Compare your answer with the correct one above

Question

Which value for proves that the function will increases faster than the function ?

Answer

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 .

Step 2: Use technology to graph .

Step 3: Compare the graphs of and .

Graphically, it appears that is the point where increases more rapidly than . Substitute this value into both functions to algebraic verify the assumption.

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

Since

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

Step 4: Answer the question.

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

Compare your answer with the correct one above

Question

Which value for proves that the function will increases faster than the function ?

Answer

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 .

Step 2: Use technology to graph .

Step 3: Compare the graphs of and .

Graphically, it appears that is the point where increases more rapidly than . Substitute this value into both functions to algebraic verify the assumption.

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

Since

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

Step 4: Answer the question.

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

Compare your answer with the correct one above

Question

Which value for proves that the function will increases faster than the function ?

Answer

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 .

Step 2: Use technology to graph .

Step 3: Compare the graphs of and .

Graphically, it appears that is the point where increases more rapidly than . Substitute this value into both functions to algebraic verify the assumption.

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

Since

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

Step 4: Answer the question.

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

Compare your answer with the correct one above

Question

Which value for proves that the function will increases faster than the function ?

Answer

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 .

Step 2: Use technology to graph .

Step 3: Compare the graphs of and .

Graphically, it appears that is the point where increases more rapidly than . Substitute this value into both functions to algebraic verify the assumption.

$f(2)=e^{2}=(2.718)^{2}=7.389$

Since

$7.389>7\Rightarrow f(2)>g(2)\Rightarrow x\geq2$

Step 4: Answer the question.

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

Compare your answer with the correct one above

Question

Which value for proves that the function will increases faster than the function ?

Answer

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 .

Step 2: Use technology to graph .

Step 3: Compare the graphs of and .

Graphically, it appears that is the point where increases more rapidly than . Substitute this value into both functions to algebraic verify the assumption.

$f(2.6)=e^{2.6}=(2.718)^{2.6}=13.464$

Since

$13.464>13\Rightarrow f(2.6)>g(2.6)\Rightarrow x\geq2.6$

Step 4: Answer the question.

For values $x\geq2.6$ , $f(x)\geq g(x)$ .

Compare your answer with the correct one above

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