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Sagot :
Let's compare and contrast the domain and range for the functions [tex]\( f(x) = 5x \)[/tex] and [tex]\( g(x) = 5^x \)[/tex] to answer Keshawn's question.
### Step-by-Step Solution:
1. Domain of [tex]\( f(x) = 5x \)[/tex]:
- The domain of a function refers to the set of all possible input values (x-values) that the function can accept.
- For the function [tex]\( f(x) = 5x \)[/tex], there are no restrictions on the input value [tex]\( x \)[/tex]. You can plug in any real number for [tex]\( x \)[/tex], and it will produce a valid output.
- Hence, the domain of [tex]\( f(x) = 5x \)[/tex] is all real numbers.
2. Domain of [tex]\( g(x) = 5^x \)[/tex]:
- Similarly, the domain of [tex]\( g(x) = 5^x \)[/tex] includes all potential input values (x-values) for which the function is defined.
- In this case, [tex]\( x \)[/tex] can also be any real number because any real number can be an exponent of [tex]\( 5 \)[/tex].
- Therefore, the domain of [tex]\( g(x) = 5^x \)[/tex] is all real numbers.
3. Range of [tex]\( f(x) = 5x \)[/tex]:
- The range of a function refers to the set of all possible output values (y-values) that the function can produce.
- For the function [tex]\( f(x) = 5x \)[/tex], multiplying 5 by any real number [tex]\( x \)[/tex] can produce any real number (positive, negative, or zero).
- Thus, the range of [tex]\( f(x) = 5x \)[/tex] is all real numbers.
4. Range of [tex]\( g(x) = 5^x \)[/tex]:
- For the function [tex]\( g(x) = 5^x \)[/tex], the output is always a positive number regardless of the exponent [tex]\( x \)[/tex].
- Specifically, [tex]\( 5^x \)[/tex] never results in zero or a negative number. As [tex]\( x \)[/tex] increases, [tex]\( 5^x \)[/tex] grows larger, and as [tex]\( x \)[/tex] decreases, [tex]\( 5^x \)[/tex] approaches 0 but never reaches it.
- Therefore, the range of [tex]\( g(x) = 5^x \)[/tex] is [tex]\( y > 0 \)[/tex].
### Statements Keshawn Could Include:
Based on the above analysis, the following statements are valid and could be included in Keshawn's explanation:
1. The domain of both functions is all real numbers.
2. The range of [tex]\( g(x) \)[/tex] is [tex]\( y > 0 \)[/tex].
### Invalid Statements:
The remaining statements are incorrect:
- The domain of [tex]\( f(x) \)[/tex] is [tex]\( x > 5 \)[/tex] – This is false because the domain is all real numbers, not restricted to [tex]\( x > 5 \)[/tex].
- The domain of [tex]\( g(x) \)[/tex] is [tex]\( x > 5 \)[/tex] – This is also false for the same reason above.
- The range of [tex]\( f(x) \)[/tex] is [tex]\( y > 0 \)[/tex] – This is false because the range of [tex]\( f(x) = 5x \)[/tex] is all real numbers, not limited to [tex]\( y > 0 \)[/tex].
Therefore, the correct options Keshawn could include in his explanation are:
- The domain of both functions is all real numbers.
- The range of [tex]\( g(x) \)[/tex] is [tex]\( y > 0 \)[/tex].
### Step-by-Step Solution:
1. Domain of [tex]\( f(x) = 5x \)[/tex]:
- The domain of a function refers to the set of all possible input values (x-values) that the function can accept.
- For the function [tex]\( f(x) = 5x \)[/tex], there are no restrictions on the input value [tex]\( x \)[/tex]. You can plug in any real number for [tex]\( x \)[/tex], and it will produce a valid output.
- Hence, the domain of [tex]\( f(x) = 5x \)[/tex] is all real numbers.
2. Domain of [tex]\( g(x) = 5^x \)[/tex]:
- Similarly, the domain of [tex]\( g(x) = 5^x \)[/tex] includes all potential input values (x-values) for which the function is defined.
- In this case, [tex]\( x \)[/tex] can also be any real number because any real number can be an exponent of [tex]\( 5 \)[/tex].
- Therefore, the domain of [tex]\( g(x) = 5^x \)[/tex] is all real numbers.
3. Range of [tex]\( f(x) = 5x \)[/tex]:
- The range of a function refers to the set of all possible output values (y-values) that the function can produce.
- For the function [tex]\( f(x) = 5x \)[/tex], multiplying 5 by any real number [tex]\( x \)[/tex] can produce any real number (positive, negative, or zero).
- Thus, the range of [tex]\( f(x) = 5x \)[/tex] is all real numbers.
4. Range of [tex]\( g(x) = 5^x \)[/tex]:
- For the function [tex]\( g(x) = 5^x \)[/tex], the output is always a positive number regardless of the exponent [tex]\( x \)[/tex].
- Specifically, [tex]\( 5^x \)[/tex] never results in zero or a negative number. As [tex]\( x \)[/tex] increases, [tex]\( 5^x \)[/tex] grows larger, and as [tex]\( x \)[/tex] decreases, [tex]\( 5^x \)[/tex] approaches 0 but never reaches it.
- Therefore, the range of [tex]\( g(x) = 5^x \)[/tex] is [tex]\( y > 0 \)[/tex].
### Statements Keshawn Could Include:
Based on the above analysis, the following statements are valid and could be included in Keshawn's explanation:
1. The domain of both functions is all real numbers.
2. The range of [tex]\( g(x) \)[/tex] is [tex]\( y > 0 \)[/tex].
### Invalid Statements:
The remaining statements are incorrect:
- The domain of [tex]\( f(x) \)[/tex] is [tex]\( x > 5 \)[/tex] – This is false because the domain is all real numbers, not restricted to [tex]\( x > 5 \)[/tex].
- The domain of [tex]\( g(x) \)[/tex] is [tex]\( x > 5 \)[/tex] – This is also false for the same reason above.
- The range of [tex]\( f(x) \)[/tex] is [tex]\( y > 0 \)[/tex] – This is false because the range of [tex]\( f(x) = 5x \)[/tex] is all real numbers, not limited to [tex]\( y > 0 \)[/tex].
Therefore, the correct options Keshawn could include in his explanation are:
- The domain of both functions is all real numbers.
- The range of [tex]\( g(x) \)[/tex] is [tex]\( y > 0 \)[/tex].
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