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Common Core: High School - Functions : High School: Functions

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

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All Common Core: High School - Functions Resources

6 Diagnostic Tests 82 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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:

$x\geq-1$

$x\geq2.2$

$x\geq0$

$x\geq1.2$

$x\geq3.1$

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.

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) .

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:

$x\geq1$

$x\geq2.7$

$x\geq3$

$x\geq0$

$x\geq2$

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:

$x\geq2$

$x\geq2.68$

$x\geq3.45$

$x\geq3.5$

$x\geq1$

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:

$x\geq3.12$

$x\geq1$

$x\leq2.8$

$x\geq2.9$

$x\geq2$

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

Example Question #11 : 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+3 ?

Possible Answers:

$x\geq3$

$x\geq1.5$

$x\geq0$

$x\geq2$

$x\geq1$

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.

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) .

Q12

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

Q2 2

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

Q12 3

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

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

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

Since

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

Step 4: Answer the question.

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

Report an Error

Example Question #12 : 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 ?

Possible Answers:

$x\geq1$

$x\geq2.6$

$x\geq6.2$

$x\geq0$

$x\geq2$

Correct answer:

$x\geq2.6$

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) .

Screen shot 2016 02 09 at 9.53.13 am

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

Q2 2

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

Screen shot 2016 02 09 at 9.53.25 am

Graphically, it appears that x=2.6 is the point where g(x) increases more rapidly than f(x) . Substitute this value into both functions to algebraic verify the assumption.

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

g(2.6)=5(2.6)=13

Since

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

Step 4: Answer the question.

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

Report an Error

Example Question #1 : Express Exponential Models As Logarithmic Solutions: Ccss.Math.Content.Hsf Le.A.4

Solve for using rules of logarithmic functions.

$e^{x-4}+2=10$

Possible Answers:

$x=\frac{\ln 8}{2}$

$x=\ln 8+4$

$x={\ln 4}$

$x=\ln 8-4$

$x=\frac{\ln 8}{4}$

Correct answer:

$x=\ln 8+4$

Explanation:

This question is testing one's ability to understand logarithmic rules and apply them in order to solve a function.

For the purpose of Common Core Standards, "For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and dare numbers and the base b is 2, 10, or e; evaluate the logarithm using technology." falls within the Cluster A of "Construct and compare linear, quadratic, and exponential models and solve problems" concept (CCSS.MATH.CONTENT.HSF-LE.A.4).

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 algebraic operations to manipulate the function and isolate the value on one side of the equation.

$e^{x-4}+2=10$

Subtract two from both sides.

$\\e^{x-4}+2-2=10-2 \\e^{x-4}=8$

Step 2: Identify logarithmic rules.

Recall that $\ln e=1$

Step 3: Apply logarithmic rules to solve for .

$\\e^{x-4}=8 \\ \ln e^{x-4}=\ln 8\\ x-4=\ln 8\\x=\ln 8+4$

Report an Error

Example Question #1 : Express Exponential Models As Logarithmic Solutions: Ccss.Math.Content.Hsf Le.A.4

Solve for using rules of logarithmic functions.

$e^{x-4}+1=5$

Possible Answers:

$x=\ln4-4$

$x=\ln4-1$

$x=\ln1+4$

$x=\ln4+4$

$x=\ln4+1$

Correct answer:

$x=\ln4+4$

Explanation:

This question is testing one's ability to understand logarithmic rules and apply them in order to solve a function.

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 algebraic operations to manipulate the function and isolate the value on one side of the equation.

$e^{x-4}+1=5$

Subtract one from both sides.

$\\e^{x-4}+1-1=5-1 \\e^{x-4}=4$

Step 2: Identify logarithmic rules.

Recall that $\ln e=1$

Step 3: Apply logarithmic rules to solve for .

$\\e^{x-4}=4 \\ \ln e^{x-4}=\ln 4\\ x-4=\ln 4\\x=\ln 4+4$

Report an Error

Example Question #3 : Express Exponential Models As Logarithmic Solutions: Ccss.Math.Content.Hsf Le.A.4

Solve for using rules of logarithmic functions.

$2e^{x}+1=5$

Possible Answers:

$x=\ln1$

$x=\ln2$

$x=\sqrt{\ln4}$

$x=\ln4$

$x=\sqrt{\ln2}$

Correct answer:

$x=\ln2$

Explanation:

This question is testing one's ability to understand logarithmic rules and apply them in order to solve a function.

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 algebraic operations to manipulate the function and isolate the value on one side of the equation.

$2e^{x}+1=5$

Subtract one from both sides.

$\\2e^{x}+1-1=5-1 \\2e^{x}=4$

Step 2: Identify logarithmic rules.

Recall that $\ln e=1$

Step 3: Apply logarithmic rules to solve for .

$\\2e^x=4 \\e^x=\frac{4}{2} \\ \\e^x=2 \\ \ln e^x=\ln2 \\x=\ln 2$

Report an Error

Example Question #2 : Express Exponential Models As Logarithmic Solutions: Ccss.Math.Content.Hsf Le.A.4

Solve for using rules of logarithmic functions.

$2e^{x}+5=17$

Possible Answers:

$x=\ln6+2$

$x=\ln4-2$

$x=\ln6$

$x=\ln4$

$x=\ln12-2$

Correct answer:

$x=\ln6$

Explanation:

This question is testing one's ability to understand logarithmic rules and apply them in order to solve a function.

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 algebraic operations to manipulate the function and isolate the value on one side of the equation.

$2e^{x}+5=17$

Subtract five from both sides and then divide by two.

$\\2e^{x}+5-5=17-5\\2e^{x}=12\\e^x=6$

Step 2: Identify logarithmic rules.

Recall that $\ln e=1$

Step 3: Apply logarithmic rules to solve for .

$\\e^x=6 \\ \ln e^x=\ln6 \\x=\ln 6$

Report an Error

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All Common Core: High School - Functions Resources

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Common Core: High School - Functions : High School: Functions

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

All Common Core: High School - Functions Resources

Example Questions

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

Example Question #11 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Example Question #12 : Exponential Functions Exceeding Polynomial Functions: Ccss.Math.Content.Hsf Le.A.3

Example Question #1 : Express Exponential Models As Logarithmic Solutions: Ccss.Math.Content.Hsf Le.A.4

Example Question #1 : Express Exponential Models As Logarithmic Solutions: Ccss.Math.Content.Hsf Le.A.4

Example Question #3 : Express Exponential Models As Logarithmic Solutions: Ccss.Math.Content.Hsf Le.A.4

Example Question #2 : Express Exponential Models As Logarithmic Solutions: Ccss.Math.Content.Hsf Le.A.4

All Common Core: High School - Functions Resources

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