Answer :
Let's examine the function [tex]\( f(x) = 5^{x-2} \)[/tex] step by step to determine its domain, range, horizontal asymptote, and [tex]\( y \)[/tex]-intercept, as well as to graph it using transformations.
### Step 1: Domain
For any exponential function of the form [tex]\( a^{(x+b)} \)[/tex], the domain includes all real numbers because the exponent [tex]\( x \)[/tex] can take any real value. Therefore, the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[ \text{Domain: } (-\infty, \infty) \][/tex]
### Step 2: Range
Exponential functions like [tex]\( 5^{x-2} \)[/tex] are always positive for any real number exponent because any positive base raised to any power is positive. Therefore, the range of [tex]\( f(x) = 5^{x-2} \)[/tex] is:
[tex]\[ \text{Range: } (0, \infty) \][/tex]
### Step 3: Horizontal Asymptote
A horizontal asymptote for an exponential function [tex]\( a^{(x+b)} \)[/tex] can be found by examining the behavior of the function as [tex]\( x \to \infty \)[/tex] and [tex]\( x \to -\infty \)[/tex]. As [tex]\( x \to -\infty \)[/tex], [tex]\( 5^{x-2} \)[/tex] approaches 0, but never actually reaches 0. Thus, the horizontal asymptote is:
[tex]\[ \text{Horizontal Asymptote: } y = 0 \][/tex]
### Step 4: [tex]\( y \)[/tex]-Intercept
To find the [tex]\( y \)[/tex]-intercept, we set [tex]\( x = 0 \)[/tex] in the function and solve for [tex]\( y \)[/tex]:
[tex]\[ f(0) = 5^{0-2} = 5^{-2} = \frac{1}{5^2} = \frac{1}{25} = 0.04 \][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept is:
[tex]\[ y \text{-intercept: } 0.04 \][/tex]
### Graphing Using Transformations
To graph [tex]\( f(x) = 5^{x-2} \)[/tex], you start with the basic exponential function [tex]\( y = 5^x \)[/tex], which has a base of 5.
1. Shift to the right by 2 units: The term [tex]\( x-2 \)[/tex] indicates a horizontal shift 2 units to the right.
Therefore, the graph of [tex]\( f(x) = 5^{x-2} \)[/tex] will be a standard exponential growth curve shifted 2 units to the right.
### Summary
- Domain: [tex]\((-\infty, \infty)\)[/tex]
- Range: [tex]\((0, \infty)\)[/tex]
- Horizontal Asymptote: [tex]\( y = 0 \)[/tex]
- [tex]\( y \)[/tex]-Intercept: [tex]\( 0.04 \)[/tex]
Let's address the multiple-choice question:
- The range of [tex]\( f(x)=5^{x-2} \)[/tex] is [tex]\( (0, \infty) \)[/tex].
- There is one horizontal asymptote, and it is [tex]\( y = 0 \)[/tex].
Thus, the correct choice is:
B. The function has one horizontal asymptote, [tex]\( y = 0 \)[/tex].
### Step 1: Domain
For any exponential function of the form [tex]\( a^{(x+b)} \)[/tex], the domain includes all real numbers because the exponent [tex]\( x \)[/tex] can take any real value. Therefore, the domain of [tex]\( f(x) \)[/tex] is:
[tex]\[ \text{Domain: } (-\infty, \infty) \][/tex]
### Step 2: Range
Exponential functions like [tex]\( 5^{x-2} \)[/tex] are always positive for any real number exponent because any positive base raised to any power is positive. Therefore, the range of [tex]\( f(x) = 5^{x-2} \)[/tex] is:
[tex]\[ \text{Range: } (0, \infty) \][/tex]
### Step 3: Horizontal Asymptote
A horizontal asymptote for an exponential function [tex]\( a^{(x+b)} \)[/tex] can be found by examining the behavior of the function as [tex]\( x \to \infty \)[/tex] and [tex]\( x \to -\infty \)[/tex]. As [tex]\( x \to -\infty \)[/tex], [tex]\( 5^{x-2} \)[/tex] approaches 0, but never actually reaches 0. Thus, the horizontal asymptote is:
[tex]\[ \text{Horizontal Asymptote: } y = 0 \][/tex]
### Step 4: [tex]\( y \)[/tex]-Intercept
To find the [tex]\( y \)[/tex]-intercept, we set [tex]\( x = 0 \)[/tex] in the function and solve for [tex]\( y \)[/tex]:
[tex]\[ f(0) = 5^{0-2} = 5^{-2} = \frac{1}{5^2} = \frac{1}{25} = 0.04 \][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept is:
[tex]\[ y \text{-intercept: } 0.04 \][/tex]
### Graphing Using Transformations
To graph [tex]\( f(x) = 5^{x-2} \)[/tex], you start with the basic exponential function [tex]\( y = 5^x \)[/tex], which has a base of 5.
1. Shift to the right by 2 units: The term [tex]\( x-2 \)[/tex] indicates a horizontal shift 2 units to the right.
Therefore, the graph of [tex]\( f(x) = 5^{x-2} \)[/tex] will be a standard exponential growth curve shifted 2 units to the right.
### Summary
- Domain: [tex]\((-\infty, \infty)\)[/tex]
- Range: [tex]\((0, \infty)\)[/tex]
- Horizontal Asymptote: [tex]\( y = 0 \)[/tex]
- [tex]\( y \)[/tex]-Intercept: [tex]\( 0.04 \)[/tex]
Let's address the multiple-choice question:
- The range of [tex]\( f(x)=5^{x-2} \)[/tex] is [tex]\( (0, \infty) \)[/tex].
- There is one horizontal asymptote, and it is [tex]\( y = 0 \)[/tex].
Thus, the correct choice is:
B. The function has one horizontal asymptote, [tex]\( y = 0 \)[/tex].
