Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Discover detailed answers to your questions from a wide network of experts on our comprehensive 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]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
