QuestionIdentify the types of functions $f(x)=x^{3}+x^{2}-3 x+4$ and $g(x)=2^{x}-4$ . What do they have in common?

Studdy Solution

STEP 1

Assumptions1. The functions are $f(x)=x^{3}+x^{}-3x+4$ and $g(x)=^{x}-4$ . . We need to identify the type of these functions and justify our answer.
3. We also need to identify the key features that these functions have in common, considering domain, range, $x$ -intercepts, and $y$ -intercepts.

STEP 2

Let's start with identifying the type of the function $f(x)=x^{}+x^{2}-x+4$ .This is a polynomial function because it is a sum of terms, where each term is a constant multiplied by a variable raised to a nonnegative integer power.

STEP 3

Now, let's identify the type of the function $g(x)=2^{x}-$ .This is an exponential function because it is a function where the variable $x$ is an exponent.

STEP 4

Now, let's find the domain of the functions $f(x)$ and $g(x)$ .
The domain of a function is the set of all possible input values (often the "x" variable), which produce a valid output from a particular function.

STEP 5

The domain of the polynomial function $f(x)$ is all real numbers, because you can plug any real number into the function and get a real number out.
So, the domain of $f(x)$ is $(-\infty, \infty)$ .

STEP 6

The domain of the exponential function $g(x)$ is also all real numbers, because you can plug any real number into the function and get a real number out.
So, the domain of $g(x)$ is $(-\infty, \infty)$ .

STEP 7

Now, let's find the range of the functions $f(x)$ and $g(x)$ .
The range of a function is the set of all possible output values (often the "y" variable), which result from the function.

STEP 8

The range of the polynomial function $f(x)$ is all real numbers, because the function can produce any real number depending on the input.
So, the range of $f(x)$ is $(-\infty, \infty)$ .

STEP 9

The range of the exponential function $g(x)$ is all real numbers greater than $-4$ , because the function can produce any real number greater than $-4$ depending on the input.
So, the range of $g(x)$ is $(-4, \infty)$ .

STEP 10

Now, let's find the $x$ -intercepts of the functions $f(x)$ and $g(x)$ .
The $x$ -intercept of a function is the point where the function crosses the x-axis.

STEP 11

The $x$ -intercepts of the function $f(x)$ are the solutions to the equation $f(x) =0$ , i.e., $x^{3}+x^{}-3x+4 =0$ .
olving this equation is not straightforward and requires knowledge of cubic equation solving techniques or numerical methods.

STEP 12

The $x$ -intercept of the function $g(x)$ is the solution to the equation $g(x) =0$ , i.e., $2^{x}-4 =0$ .
olving this equation gives $x = \log_{2}(4) =2$ .

STEP 13

Now, let's find the $y$ -intercepts of the functions $f(x)$ and $g(x)$ .
The $y$ -intercept of a function is the point where the function crosses the y-axis.

STEP 14

The $y$ -intercept of the function $f(x)$ is the value of the function at $x =0$ , i.e., $f(0) =0^{3}+0^{2}-3(0)+4 =4$ .
So, the $y$ -intercept of $f(x)$ is $4$ .

STEP 15

The $y$ -intercept of the function $g(x)$ is the value of the function at $x =0$ , i.e., $g(0) =2^{0}-4 =-4 = -3$ .
So, the $y$ -intercept of $g(x)$ is $-3$ .

STEP 16

Now, let's identify the key features that the functions $f(x)$ and $g(x)$ have in common.
From the above analysis, we can see that the only common feature between the functions $f(x)$ and $g(x)$ is their domain, which is all real numbers.

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QuestionIdentify the types of functions $f(x)=x^{3}+x^{2}-3 x+4$ and $g(x)=2^{x}-4$ . What do they have in common?

Studdy Solution

STEP 1

Assumptions1. The functions are $f(x)=x^{3}+x^{}-3x+4$ and $g(x)=^{x}-4$ . . We need to identify the type of these functions and justify our answer.
3. We also need to identify the key features that these functions have in common, considering domain, range, $x$ -intercepts, and $y$ -intercepts.

STEP 2

Let's start with identifying the type of the function $f(x)=x^{}+x^{2}-x+4$ .This is a polynomial function because it is a sum of terms, where each term is a constant multiplied by a variable raised to a nonnegative integer power.

STEP 3

Now, let's identify the type of the function $g(x)=2^{x}-$ .This is an exponential function because it is a function where the variable $x$ is an exponent.

STEP 4

Now, let's find the domain of the functions $f(x)$ and $g(x)$ .
The domain of a function is the set of all possible input values (often the "x" variable), which produce a valid output from a particular function.

STEP 5

The domain of the polynomial function $f(x)$ is all real numbers, because you can plug any real number into the function and get a real number out.
So, the domain of $f(x)$ is $(-\infty, \infty)$ .

STEP 6

The domain of the exponential function $g(x)$ is also all real numbers, because you can plug any real number into the function and get a real number out.
So, the domain of $g(x)$ is $(-\infty, \infty)$ .

STEP 7

Now, let's find the range of the functions $f(x)$ and $g(x)$ .
The range of a function is the set of all possible output values (often the "y" variable), which result from the function.

STEP 8

The range of the polynomial function $f(x)$ is all real numbers, because the function can produce any real number depending on the input.
So, the range of $f(x)$ is $(-\infty, \infty)$ .

STEP 9

The range of the exponential function $g(x)$ is all real numbers greater than $-4$ , because the function can produce any real number greater than $-4$ depending on the input.
So, the range of $g(x)$ is $(-4, \infty)$ .

STEP 10

Now, let's find the $x$ -intercepts of the functions $f(x)$ and $g(x)$ .
The $x$ -intercept of a function is the point where the function crosses the x-axis.

STEP 11

The $x$ -intercepts of the function $f(x)$ are the solutions to the equation $f(x) =0$ , i.e., $x^{3}+x^{}-3x+4 =0$ .
olving this equation is not straightforward and requires knowledge of cubic equation solving techniques or numerical methods.

STEP 12

The $x$ -intercept of the function $g(x)$ is the solution to the equation $g(x) =0$ , i.e., $2^{x}-4 =0$ .
olving this equation gives $x = \log_{2}(4) =2$ .

STEP 13

Now, let's find the $y$ -intercepts of the functions $f(x)$ and $g(x)$ .
The $y$ -intercept of a function is the point where the function crosses the y-axis.

STEP 14

The $y$ -intercept of the function $f(x)$ is the value of the function at $x =0$ , i.e., $f(0) =0^{3}+0^{2}-3(0)+4 =4$ .
So, the $y$ -intercept of $f(x)$ is $4$ .

STEP 15

The $y$ -intercept of the function $g(x)$ is the value of the function at $x =0$ , i.e., $g(0) =2^{0}-4 =-4 = -3$ .
So, the $y$ -intercept of $g(x)$ is $-3$ .

STEP 16

Now, let's identify the key features that the functions $f(x)$ and $g(x)$ have in common.
From the above analysis, we can see that the only common feature between the functions $f(x)$ and $g(x)$ is their domain, which is all real numbers.

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QuestionIdentify the types of functions f(x)=x3+x2−3x+4f(x)=x^{3}+x^{2}-3 x+4f(x)=x3+x2−3x+4 and g(x)=2x−4g(x)=2^{x}-4g(x)=2x−4. What do they have in common?

Studdy Solution

STEP 1

STEP 2

Let's start with identifying the type of the function f(x)=x+x2−x+4f(x)=x^{}+x^{2}-x+4f(x)=x+x2−x+4.This is a polynomial function because it is a sum of terms, where each term is a constant multiplied by a variable raised to a nonnegative integer power.

STEP 3

Now, let's identify the type of the function g(x)=2x−g(x)=2^{x}-g(x)=2x−.This is an exponential function because it is a function where the variable xxx is an exponent.

STEP 4

Now, let's find the domain of the functions f(x)f(x)f(x) and g(x)g(x)g(x).The domain of a function is the set of all possible input values (often the "x" variable), which produce a valid output from a particular function.

STEP 5

The domain of the polynomial function f(x)f(x)f(x) is all real numbers, because you can plug any real number into the function and get a real number out.So, the domain of f(x)f(x)f(x) is (−∞,∞)(-\infty, \infty)(−∞,∞).

STEP 6

The domain of the exponential function g(x)g(x)g(x) is also all real numbers, because you can plug any real number into the function and get a real number out.So, the domain of g(x)g(x)g(x) is (−∞,∞)(-\infty, \infty)(−∞,∞).

STEP 7

Now, let's find the range of the functions f(x)f(x)f(x) and g(x)g(x)g(x).The range of a function is the set of all possible output values (often the "y" variable), which result from the function.

STEP 8

The range of the polynomial function f(x)f(x)f(x) is all real numbers, because the function can produce any real number depending on the input.So, the range of f(x)f(x)f(x) is (−∞,∞)(-\infty, \infty)(−∞,∞).

STEP 9

The range of the exponential function g(x)g(x)g(x) is all real numbers greater than −4-4−4, because the function can produce any real number greater than −4-4−4 depending on the input.So, the range of g(x)g(x)g(x) is (−4,∞)(-4, \infty)(−4,∞).

STEP 10

Now, let's find the xxx-intercepts of the functions f(x)f(x)f(x) and g(x)g(x)g(x).The xxx-intercept of a function is the point where the function crosses the x-axis.

STEP 11

The xxx-intercepts of the function f(x)f(x)f(x) are the solutions to the equation f(x)=0f(x) =0f(x)=0, i.e., x3+x−3x+4=0x^{3}+x^{}-3x+4 =0x3+x−3x+4=0.olving this equation is not straightforward and requires knowledge of cubic equation solving techniques or numerical methods.

STEP 12

The xxx-intercept of the function g(x)g(x)g(x) is the solution to the equation g(x)=0g(x) =0g(x)=0, i.e., 2x−4=02^{x}-4 =02x−4=0.olving this equation gives x=log⁡2(4)=2x = \log_{2}(4) =2x=log2​(4)=2.

STEP 13

Now, let's find the yyy-intercepts of the functions f(x)f(x)f(x) and g(x)g(x)g(x).The yyy-intercept of a function is the point where the function crosses the y-axis.

STEP 14

The yyy-intercept of the function f(x)f(x)f(x) is the value of the function at x=0x =0x=0, i.e., f(0)=03+02−3(0)+4=4f(0) =0^{3}+0^{2}-3(0)+4 =4f(0)=03+02−3(0)+4=4.So, the yyy-intercept of f(x)f(x)f(x) is 444.

STEP 15

The yyy-intercept of the function g(x)g(x)g(x) is the value of the function at x=0x =0x=0, i.e., g(0)=20−4=−4=−3g(0) =2^{0}-4 =-4 = -3g(0)=20−4=−4=−3.So, the yyy-intercept of g(x)g(x)g(x) is −3-3−3.

STEP 16

Now, let's identify the key features that the functions f(x)f(x)f(x) and g(x)g(x)g(x) have in common.From the above analysis, we can see that the only common feature between the functions f(x)f(x)f(x) and g(x)g(x)g(x) is their domain, which is all real numbers.

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QuestionIdentify the types of functions $f(x)=x^{3}+x^{2}-3 x+4$ and $g(x)=2^{x}-4$ . What do they have in common?

Let's start with identifying the type of the function $f(x)=x^{}+x^{2}-x+4$ .This is a polynomial function because it is a sum of terms, where each term is a constant multiplied by a variable raised to a nonnegative integer power.

Now, let's identify the type of the function $g(x)=2^{x}-$ .This is an exponential function because it is a function where the variable $x$ is an exponent.

Now, let's find the domain of the functions $f(x)$ and $g(x)$ .
The domain of a function is the set of all possible input values (often the "x" variable), which produce a valid output from a particular function.

The domain of the polynomial function $f(x)$ is all real numbers, because you can plug any real number into the function and get a real number out.
So, the domain of $f(x)$ is $(-\infty, \infty)$ .

The domain of the exponential function $g(x)$ is also all real numbers, because you can plug any real number into the function and get a real number out.
So, the domain of $g(x)$ is $(-\infty, \infty)$ .

Now, let's find the range of the functions $f(x)$ and $g(x)$ .
The range of a function is the set of all possible output values (often the "y" variable), which result from the function.

The range of the polynomial function $f(x)$ is all real numbers, because the function can produce any real number depending on the input.
So, the range of $f(x)$ is $(-\infty, \infty)$ .

The range of the exponential function $g(x)$ is all real numbers greater than $-4$ , because the function can produce any real number greater than $-4$ depending on the input.
So, the range of $g(x)$ is $(-4, \infty)$ .

Now, let's find the $x$ -intercepts of the functions $f(x)$ and $g(x)$ .
The $x$ -intercept of a function is the point where the function crosses the x-axis.

The $x$ -intercepts of the function $f(x)$ are the solutions to the equation $f(x) =0$ , i.e., $x^{3}+x^{}-3x+4 =0$ .
olving this equation is not straightforward and requires knowledge of cubic equation solving techniques or numerical methods.

The $x$ -intercept of the function $g(x)$ is the solution to the equation $g(x) =0$ , i.e., $2^{x}-4 =0$ .
olving this equation gives $x = \log_{2}(4) =2$ .

Now, let's find the $y$ -intercepts of the functions $f(x)$ and $g(x)$ .
The $y$ -intercept of a function is the point where the function crosses the y-axis.

The $y$ -intercept of the function $f(x)$ is the value of the function at $x =0$ , i.e., $f(0) =0^{3}+0^{2}-3(0)+4 =4$ .
So, the $y$ -intercept of $f(x)$ is $4$ .

The $y$ -intercept of the function $g(x)$ is the value of the function at $x =0$ , i.e., $g(0) =2^{0}-4 =-4 = -3$ .
So, the $y$ -intercept of $g(x)$ is $-3$ .

Now, let's identify the key features that the functions $f(x)$ and $g(x)$ have in common.
From the above analysis, we can see that the only common feature between the functions $f(x)$ and $g(x)$ is their domain, which is all real numbers.