Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To determine which functions could be exponential, we will analyze the given set of values for [tex]\( f(x) \)[/tex], [tex]\( g(x) \)[/tex], [tex]\( h(x) \)[/tex], and [tex]\( k(x) \)[/tex] to see if they follow the common pattern of exponential functions.
An exponential function has the form [tex]\( y = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants, and for any two consecutive points, the ratio of [tex]\( y \)[/tex]-values should be constant.
Let’s analyze each function step by step:
### Function [tex]\( f(x) \)[/tex]:
[tex]\[ \begin{aligned} &x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ &f(x) & 1 & 4 & 9 & 16 & 25 & 36 & 49 \\ \end{aligned} \][/tex]
Let’s calculate the ratios of consecutive [tex]\( y \)[/tex]-values:
[tex]\[ \frac{4}{1} = 4, \quad \frac{9}{4} = 2.25, \quad \frac{16}{9} \approx 1.78, \quad \frac{25}{16} \approx 1.56, \quad \frac{36}{25} = 1.44, \quad \frac{49}{36} \approx 1.36 \][/tex]
The ratios are not constant. Thus, [tex]\( f(x) \)[/tex] is not an exponential function.
### Function [tex]\( g(x) \)[/tex]:
[tex]\[ \begin{aligned} &x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ &g(x) & 0.125 & 0.25 & 0.5 & 1 & 2 & 4 & 8 \\ \end{aligned} \][/tex]
Let’s calculate the ratios of consecutive [tex]\( y \)[/tex]-values:
[tex]\[ \frac{0.25}{0.125} = 2, \quad \frac{0.5}{0.25} = 2, \quad \frac{1}{0.5} = 2, \quad \frac{2}{1} = 2, \quad \frac{4}{2} = 2, \quad \frac{8}{4} = 2 \][/tex]
The ratios are constant. Thus, [tex]\( g(x) \)[/tex] could be an exponential function.
### Function [tex]\( h(x) \)[/tex]:
[tex]\[ \begin{aligned} &x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ &h(x) & 0.25 & 0.5 & 0.75 & 0 & 1.25 & 1.5 & 1.75 \\ \end{aligned} \][/tex]
Let’s calculate the ratios of consecutive [tex]\( y \)[/tex]-values:
[tex]\[ \frac{0.5}{0.25} = 2, \quad \frac{0.75}{0.5} = 1.5, \quad \frac{0}{0.75} = 0, \quad \frac{1.25}{0} \ (\text{undefined}), \quad \frac{1.5}{1.25} = 1.2, \quad \frac{1.75}{1.5} \approx 1.17 \][/tex]
The ratios are not constant. Thus, [tex]\( h(x) \)[/tex] is not an exponential function.
### Function [tex]\( k(x) \)[/tex]:
[tex]\[ \begin{aligned} &x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ &k(x) & 64 & 16 & 4 & 1 & 0.25 & 0.0625 & 0.0156 \\ \end{aligned} \][/tex]
Let’s calculate the ratios of consecutive [tex]\( y \)[/tex]-values:
[tex]\[ \frac{16}{64} = 0.25, \quad \frac{4}{16} = 0.25, \quad \frac{1}{4} = 0.25, \quad \frac{0.25}{1} = 0.25, \quad \frac{0.0625}{0.25} = 0.25, \quad \frac{0.0156}{0.0625} = 0.25 \][/tex]
The ratios are constant. Thus, [tex]\( k(x) \)[/tex] could be an exponential function.
### Conclusion:
The functions [tex]\( g(x) \)[/tex] and [tex]\( k(x) \)[/tex] give values that could come from exponential functions.
So, the correct answers are:
- A. [tex]\( g(x) \)[/tex]
- D. [tex]\( k(x) \)[/tex]
An exponential function has the form [tex]\( y = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants, and for any two consecutive points, the ratio of [tex]\( y \)[/tex]-values should be constant.
Let’s analyze each function step by step:
### Function [tex]\( f(x) \)[/tex]:
[tex]\[ \begin{aligned} &x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ &f(x) & 1 & 4 & 9 & 16 & 25 & 36 & 49 \\ \end{aligned} \][/tex]
Let’s calculate the ratios of consecutive [tex]\( y \)[/tex]-values:
[tex]\[ \frac{4}{1} = 4, \quad \frac{9}{4} = 2.25, \quad \frac{16}{9} \approx 1.78, \quad \frac{25}{16} \approx 1.56, \quad \frac{36}{25} = 1.44, \quad \frac{49}{36} \approx 1.36 \][/tex]
The ratios are not constant. Thus, [tex]\( f(x) \)[/tex] is not an exponential function.
### Function [tex]\( g(x) \)[/tex]:
[tex]\[ \begin{aligned} &x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ &g(x) & 0.125 & 0.25 & 0.5 & 1 & 2 & 4 & 8 \\ \end{aligned} \][/tex]
Let’s calculate the ratios of consecutive [tex]\( y \)[/tex]-values:
[tex]\[ \frac{0.25}{0.125} = 2, \quad \frac{0.5}{0.25} = 2, \quad \frac{1}{0.5} = 2, \quad \frac{2}{1} = 2, \quad \frac{4}{2} = 2, \quad \frac{8}{4} = 2 \][/tex]
The ratios are constant. Thus, [tex]\( g(x) \)[/tex] could be an exponential function.
### Function [tex]\( h(x) \)[/tex]:
[tex]\[ \begin{aligned} &x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ &h(x) & 0.25 & 0.5 & 0.75 & 0 & 1.25 & 1.5 & 1.75 \\ \end{aligned} \][/tex]
Let’s calculate the ratios of consecutive [tex]\( y \)[/tex]-values:
[tex]\[ \frac{0.5}{0.25} = 2, \quad \frac{0.75}{0.5} = 1.5, \quad \frac{0}{0.75} = 0, \quad \frac{1.25}{0} \ (\text{undefined}), \quad \frac{1.5}{1.25} = 1.2, \quad \frac{1.75}{1.5} \approx 1.17 \][/tex]
The ratios are not constant. Thus, [tex]\( h(x) \)[/tex] is not an exponential function.
### Function [tex]\( k(x) \)[/tex]:
[tex]\[ \begin{aligned} &x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ &k(x) & 64 & 16 & 4 & 1 & 0.25 & 0.0625 & 0.0156 \\ \end{aligned} \][/tex]
Let’s calculate the ratios of consecutive [tex]\( y \)[/tex]-values:
[tex]\[ \frac{16}{64} = 0.25, \quad \frac{4}{16} = 0.25, \quad \frac{1}{4} = 0.25, \quad \frac{0.25}{1} = 0.25, \quad \frac{0.0625}{0.25} = 0.25, \quad \frac{0.0156}{0.0625} = 0.25 \][/tex]
The ratios are constant. Thus, [tex]\( k(x) \)[/tex] could be an exponential function.
### Conclusion:
The functions [tex]\( g(x) \)[/tex] and [tex]\( k(x) \)[/tex] give values that could come from exponential functions.
So, the correct answers are:
- A. [tex]\( g(x) \)[/tex]
- D. [tex]\( k(x) \)[/tex]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.
