{"id":1250,"date":"2015-11-12T18:35:30","date_gmt":"2015-11-12T18:35:30","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1250"},"modified":"2017-04-03T18:49:54","modified_gmt":"2017-04-03T18:49:54","slug":"identify-power-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/identify-power-functions\/","title":{"raw":"Identify power functions","rendered":"Identify power functions"},"content":{"raw":"
In order to better understand the bird problem, we need to understand a specific type of function. A power function <\/strong>is a function with a single term that is the product of a real number, a coefficient,<\/strong> and a variable raised to a fixed real number. (A number that multiplies a variable raised to an exponent is known as a coefficient.) As an example, consider functions for area or volume. The function for the area of a circle<\/strong> with radius [latex]r[\/latex] is\r\n
$$A \\left(r\\right)=\\pi {r}^{2}\\$$<\/div>\r\nand the function for the volume of a sphere<\/strong> with radius [latex]r[\/latex]\u00a0is\r\n
$$V \\left(r\\right)=\\frac{4}{3}\\pi {r}^{3}\\$$<\/div>\r\nBoth of these are examples of power functions because they consist of a coefficient, [latex]\\pi [\/latex] or [latex]\\frac{4}{3}\\pi [\/latex], multiplied by a variable [latex]r[\/latex]\u00a0raised to a power.\r\n
\r\n

A General Note: Power Function<\/h3>\r\nA power function<\/strong> is a function that can be represented in the form\r\n
[latex]f\\left(x\\right)=k{x}^{p}[\/latex]<\/div>\r\nwhere k<\/em>\u00a0and p<\/em>\u00a0are real numbers, and k<\/em>\u00a0is known as the coefficient<\/strong>.\r\n\r\n<\/div>\r\n
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Q & A<\/h3>\r\nIs [latex]f\\left(x\\right)={2}^{x}[\/latex] a power function?<\/strong>\r\n\r\nNo. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function.<\/em>\r\n\r\n<\/div>\r\n
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Example 1: Identifying Power Functions<\/h3>\r\nWhich of the following functions are power functions?\r\n

[latex]begin{cases}f\\left(x\\right)=1hfill & text{Constant function}hfill \\ f\\left(x\\right)=xhfill & text{Identify function}hfill \\ f\\left(x\\right)={x}^{2}hfill & text{Quadratic}text{ }text{ function}hfill \\ f\\left(x\\right)={x}^{3}hfill & text{Cubic function}hfill \\ f\\left(x\\right)=\\frac{1}{x} hfill & text{Reciprocal function}hfill \\ f\\left(x\\right)=\\frac{1}{{x}^{2}}hfill & text{Reciprocal squared function}hfill \\ f\\left(x\\right)=sqrt{x}hfill & text{Square root function}hfill \\ f\\left(x\\right)=sqrt[3]{x}hfill & text{Cube root function}hfill end{cases}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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Solution<\/h3>\r\nAll of the listed functions are power functions.\r\n\r\nThe constant and identity functions are power functions because they can be written as [latex]f\\left(x\\right)={x}^{0}[\/latex] and [latex]f\\left(x\\right)={x}^{1}[\/latex] respectively.\r\n\r\nThe quadratic and cubic functions are power functions with whole number powers [latex]f\\left(x\\right)={x}^{2}[\/latex] and [latex]f\\left(x\\right)={x}^{3}[\/latex].\r\n\r\nThe reciprocal<\/strong> and reciprocal squared functions are power functions with negative whole number powers because they can be written as [latex]f\\left(x\\right)={x}^{-1}[\/latex] and [latex]f\\left(x\\right)={x}^{-2}[\/latex].\r\n\r\nThe square and cube root<\/strong> functions are power functions with \\fractional powers because they can be written as [latex]f\\left(x\\right)={x}^{1\/2}[\/latex] or [latex]f\\left(x\\right)={x}^{1\/3}[\/latex].\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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Try It 1<\/h3>\r\nWhich functions are power functions?\r\n\r\n[latex]f\\left(x\\right)=2{x}^{2}\\cdot4{x}^{3}[\/latex]\r\n\r\n[latex]g\\left(x\\right)=-{x}^{5}+5{x}^{3}-4x[\/latex]\r\n\r\n[latex]h\\left(x\\right)=\\frac{2{x}^{5}-1}{3{x}^{2}+4}[\/latex]\r\nSolution<\/a>\r\n\r\n<\/div>\r\n<\/section>
<\/section>","rendered":"
In order to better understand the bird problem, we need to understand a specific type of function. A power function <\/strong>is a function with a single term that is the product of a real number, a coefficient,<\/strong> and a variable raised to a fixed real number. (A number that multiplies a variable raised to an exponent is known as a coefficient.) As an example, consider functions for area or volume. The function for the area of a circle<\/strong> with radius [latex]r[\/latex] is<\/p>\n
$$A \\left(r\\right)=\\pi {r}^{2}\\$$<\/div>\n

and the function for the volume of a sphere<\/strong> with radius [latex]r[\/latex]\u00a0is<\/p>\n

$$V \\left(r\\right)=\\frac{4}{3}\\pi {r}^{3}\\$$<\/div>\n

Both of these are examples of power functions because they consist of a coefficient, [latex]\\pi[\/latex] or [latex]\\frac{4}{3}\\pi[\/latex], multiplied by a variable [latex]r[\/latex]\u00a0raised to a power.<\/p>\n

\n

A General Note: Power Function<\/h3>\n

A power function<\/strong> is a function that can be represented in the form<\/p>\n

[latex]f\\left(x\\right)=k{x}^{p}[\/latex]<\/div>\n

where k<\/em>\u00a0and p<\/em>\u00a0are real numbers, and k<\/em>\u00a0is known as the coefficient<\/strong>.<\/p>\n<\/div>\n

\n

Q & A<\/h3>\n

Is [latex]f\\left(x\\right)={2}^{x}[\/latex] a power function?<\/strong><\/p>\n

No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function.<\/em><\/p>\n<\/div>\n

\n
\n
\n

Example 1: Identifying Power Functions<\/h3>\n

Which of the following functions are power functions?<\/p>\n

[latex]begin{cases}f\\left(x\\right)=1hfill & text{Constant function}hfill \\ f\\left(x\\right)=xhfill & text{Identify function}hfill \\ f\\left(x\\right)={x}^{2}hfill & text{Quadratic}text{ }text{ function}hfill \\ f\\left(x\\right)={x}^{3}hfill & text{Cubic function}hfill \\ f\\left(x\\right)=\\frac{1}{x} hfill & text{Reciprocal function}hfill \\ f\\left(x\\right)=\\frac{1}{{x}^{2}}hfill & text{Reciprocal squared function}hfill \\ f\\left(x\\right)=sqrt{x}hfill & text{Square root function}hfill \\ f\\left(x\\right)=sqrt[3]{x}hfill & text{Cube root function}hfill end{cases}[\/latex]<\/p>\n<\/div>\n

\n

Solution<\/h3>\n

All of the listed functions are power functions.<\/p>\n

The constant and identity functions are power functions because they can be written as [latex]f\\left(x\\right)={x}^{0}[\/latex] and [latex]f\\left(x\\right)={x}^{1}[\/latex] respectively.<\/p>\n

The quadratic and cubic functions are power functions with whole number powers [latex]f\\left(x\\right)={x}^{2}[\/latex] and [latex]f\\left(x\\right)={x}^{3}[\/latex].<\/p>\n

The reciprocal<\/strong> and reciprocal squared functions are power functions with negative whole number powers because they can be written as [latex]f\\left(x\\right)={x}^{-1}[\/latex] and [latex]f\\left(x\\right)={x}^{-2}[\/latex].<\/p>\n

The square and cube root<\/strong> functions are power functions with \\fractional powers because they can be written as [latex]f\\left(x\\right)={x}^{1\/2}[\/latex] or [latex]f\\left(x\\right)={x}^{1\/3}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n

\n

Try It 1<\/h3>\n

Which functions are power functions?<\/p>\n

[latex]f\\left(x\\right)=2{x}^{2}\\cdot4{x}^{3}[\/latex]<\/p>\n

[latex]g\\left(x\\right)=-{x}^{5}+5{x}^{3}-4x[\/latex]<\/p>\n

[latex]h\\left(x\\right)=\\frac{2{x}^{5}-1}{3{x}^{2}+4}[\/latex]
\nSolution<\/a><\/p>\n<\/div>\n<\/section>\n

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