4. Draw a sketch graph of $$ f(x)=\left(\frac{1}{3}\right)^{x} $$. 4.1 Give the domain and range of $$ f $$. 4.2 For which values of $$ x $$ is $$ f(x)0 $$ ? 4.3 Give the coordinates of any three points on $$ f $$. 4.4 Give the equation of $$ g $$ if $$ g $$ is formed when the graph of $$ f $$ is moved 3 units down. 4.5 Give the equation of $$ h $$ if $$ h $$ is formed when the graph of $$ f $$ is moved 3 units left. 4.6 Give the equation of $$ k $$ if $$ k $$ is formed when the graph of $$ f $$ is moved 2 units upwards and 3 un to the right. 4.7 Give the domain and range of $$ g $$. 4.8 Give the domain and range of $$ h $$. 4.9 Give the domain and range of $$ k $$. 4.10 Give the asymptotes of $$ f, g, h $$ and $$ k $$.

Question

4. Draw a sketch graph of $$ f(x)=\left(\frac{1}{3}\right)^{x} $$. 4.1 Give the domain and range of $$ f $$. 4.2 For which values of $$ x $$ is $$ f(x)>0 $$ ? 4.3 Give the coordinates of any three points on $$ f $$. 4.4 Give the equation of $$ g $$ if $$ g $$ is formed when the graph of $$ f $$ is moved 3 units down. 4.5 Give the equation of $$ h $$ if $$ h $$ is formed when the graph of $$ f $$ is moved 3 units left. 4.6 Give the equation of $$ k $$ if $$ k $$ is formed when the graph of $$ f $$ is moved 2 units upwards and 3 un to the right. 4.7 Give the domain and range of $$ g $$. 4.8 Give the domain and range of $$ h $$. 4.9 Give the domain and range of $$ k $$. 4.10 Give the asymptotes of $$ f, g, h $$ and $$ k $$.

UpStudy AI · Accepted Answer

Function by following steps:
- step0: Find the y-intercept:
  $$g\left(xight)=3^{-x}-3$$
- step1: Set $$x$$=0$$:$$
  $$g\left(0ight)=3^{-0}-3$$
- step2: Evaluate:
  $$g\left(0ight)=3^{0}-3$$
- step3: Evaluate the power:
  $$g\left(0ight)=1-3$$
- step4: Subtract the numbers:
  $$g\left(0ight)=-2$$
Analyze the y intercept of the function $$ k(x)=(\frac{1}{3})^{x-3}+2 $$
Function by following steps:
- step0: Find the y-intercept:
  $$k\left(xight)=3^{-x+3}+2$$
- step1: Set $$x$$=0$$:$$
  $$k\left(0ight)=3^{-0+3}+2$$
- step2: Evaluate:
  $$k\left(0ight)=3^{0+3}+2$$
- step3: Simplify:
  $$k\left(0ight)=29$$
Analyze the y intercept of the function $$ f(x)=(\frac{1}{3})^{x} $$
Function by following steps:
- step0: Find the y-intercept:
  $$f\left(xight)=\left(\frac{1}{3}ight)^{x}$$
- step1: Set $$x$$=0$$:$$
  $$f\left(0ight)=\left(\frac{1}{3}ight)^{0}$$
- step2: Evaluate the power:
  $$f\left(0ight)=1$$
Analyze the y intercept of the function $$ h(x)=(\frac{1}{3})^{x+3} $$
Function by following steps:
- step0: Find the y-intercept:
  $$h\left(xight)=\left(\frac{1}{3}ight)^{x+3}$$
- step1: Set $$x$$=0$$:$$
  $$h\left(0ight)=\left(\frac{1}{3}ight)^{0+3}$$
- step2: Remove 0:
  $$h\left(0ight)=\left(\frac{1}{3}ight)^{3}$$
- step3: Simplify:
  $$h\left(0ight)=\frac{1}{27}$$
Find the horizontal asymptotes of $$ f(x)=(\frac{1}{3})^{x} $$.
Function by following steps:
- step0: Find the horizontal asymptotes:
  $$f\left(xight)=\left(\frac{1}{3}ight)^{x}$$
- step1: Find the horizontal asymptotes:
  $$	extrm{No horizontal asymptotes}$$
Find the horizontal asymptotes of $$ k(x)=(\frac{1}{3})^{x-3}+2 $$.
Function by following steps:
- step0: Find the horizontal asymptotes:
  $$k\left(xight)=3^{-x+3}+2$$
- step1: Find the horizontal asymptotes:
  $$	extrm{No horizontal asymptotes}$$
Find the horizontal asymptotes of $$ h(x)=(\frac{1}{3})^{x+3} $$.
Function by following steps:
- step0: Find the horizontal asymptotes:
  $$h\left(xight)=\left(\frac{1}{3}ight)^{x+3}$$
- step1: Find the horizontal asymptotes:
  $$	extrm{No horizontal asymptotes}$$
Find the horizontal asymptotes of $$ g(x)=(\frac{1}{3})^{x}-3 $$.
Function by following steps:
- step0: Find the horizontal asymptotes:
  $$g\left(xight)=3^{-x}-3$$
- step1: Find the horizontal asymptotes:
  $$	extrm{No horizontal asymptotes}$$
Let's solve the problem step by step.

### 4. Sketch Graph of $$ f(x) = \left(\frac{1}{3}ight)^{x} $$

1. **Domain and Range of $$ f $$**:
   - The function $$ f(x) = \left(\frac{1}{3}ight)^{x} $$ is defined for all real numbers $$ x $$.
   - As $$ x $$ approaches infinity, $$ f(x) $$ approaches 0 but never reaches it. As $$ x $$ approaches negative infinity, $$ f(x) $$ approaches infinity.
   - Therefore, the domain is $$ (-\infty, \infty) $$ and the range is $$ (0, \infty) $$.

2. **Values of $$ x $$ for which $$ f(x) > 0 $$**:
   - Since $$ f(x) = \left(\frac{1}{3}ight)^{x} $$ is an exponential function, it is always positive for all real $$ x $$.
   - Thus, $$ f(x) > 0 $$ for all $$ x \in (-\infty, \infty) $$.

3. **Coordinates of Three Points on $$ f $$**:
   - Let's calculate three points:
     - $$ f(0) = \left(\frac{1}{3}ight)^{0} = 1 $$ → Point: $$ (0, 1) $$
     - $$ f(1) = \left(\frac{1}{3}ight)^{1} = \frac{1}{3} $$ → Point: $$ (1, \frac{1}{3}) $$
     - $$ f(-1) = \left(\frac{1}{3}ight)^{-1} = 3 $$ → Point: $$ (-1, 3) $$

4. **Equation of $$ g $$** (moved 3 units down):
   - The equation of $$ g $$ is $$ g(x) = f(x) - 3 = \left(\frac{1}{3}ight)^{x} - 3 $$.

5. **Equation of $$ h $$** (moved 3 units left):
   - The equation of $$ h $$ is $$ h(x) = f(x + 3) = \left(\frac{1}{3}ight)^{x + 3} = \left(\frac{1}{3}ight)^{x} \cdot \left(\frac{1}{3}ight)^{3} = \frac{1}{27} \left(\frac{1}{3}ight)^{x} $$.

6. **Equation of $$ k $$** (moved 2 units upwards and 3 units to the right):
   - The equation of $$ k $$ is $$ k(x) = f(x - 3) + 2 = \left(\frac{1}{3}ight)^{x - 3} + 2 $$.

7. **Domain and Range of $$ g $$**:
   - The domain of $$ g $$ is the same as $$ f $$: $$ (-\infty, \infty) $$.
   - The range of $$ g $$ is $$ (-3, \infty) $$ since it shifts down by 3.

8. **Domain and Range of $$ h $$**:
   - The domain of $$ h $$ is $$ (-\infty, \infty) $$.
   - The range of $$ h $$ is $$ (0, \frac{1}{27}) $$ since it is a scaled version of $$ f $$.

9. **Domain and Range of $$ k $$**:
   - The domain of $$ k $$ is $$ (-\infty, \infty) $$.
   - The range of $$ k $$ is $$ (2, \infty) $$ since it shifts up by 2.

10. **Asymptotes of $$ f, g, h, $$ and $$ k $$**:
    - All functions have a horizontal asymptote at $$ y = 0 $$ since they approach 0 as $$ x $$ approaches infinity.

### Summary of Results:
- **Domain of $$ f $$**: $$ (-\infty, \infty) $$
- **Range of $$ f $$**: $$ (0, \infty) $$
- **Values of $$ x $$ for $$ f(x) > 0 $$**: All $$ x \in (-\infty, \infty) $$
- **Three Points on $$ f $$**: $$ (0, 1), (1, \frac{1}{3}), (-1, 3) $$
- **Equation of $$ g $$**: $$ g(x) = \left(\frac{1}{3}ight)^{x} - 3 $$
- **Equation of $$ h $$**: $$ h(x) = \left(\frac{1}{3}ight)^{x + 3} $$
- **Equation of $$ k $$**: $$ k(x) = \left(\frac{1}{3}ight)^{x - 3} + 2 $$
- **Domain of $$ g $$**: $$ (-\infty, \infty) $$
- **Range of $$ g $$**: $$ (-3, \infty) $$
- **Domain of $$ h $$**: $$ (-\infty, \infty) $$
- **Range of $$ h $$**: $$ (0, \frac{1}{27}) $$
- **Domain of $$ k $$**: $$ (-\infty, \infty) $$
- **Range of $$ k $$**: $$ (2, \infty) $$
- **Asymptotes**: All functions have a horizontal asymptote at $$ y = 0 $$.
**Summary of Results:** 1. **Function $$ f(x) = \left(\frac{1}{3}ight)^{x} $$** - **Domain**: All real numbers ($$ (-\infty, \infty) $$) - **Range**: Positive real numbers ($$ (0, \infty) $$) - **$$ f(x) > 0 $$**: For all real $$ x $$ - **Three Points**: $$ (0, 1) $$, $$ (1, \frac{1}{3}) $$, $$ (-1, 3) $$ 2. **Transformed Functions:** - **$$ g(x) $$**: $$ g(x) = \left(\frac{1}{3}ight)^{x} - 3 $$ - **$$ h(x) $$**: $$ h(x) = \left(\frac{1}{3}ight)^{x + 3} $$ - **$$ k(x) $$**: $$ k(x) = \left(\frac{1}{3}ight)^{x - 3} + 2 $$ 3. **Properties of Transformed Functions:** - **$$ g(x) $$** - **Domain**: All real numbers ($$ (-\infty, \infty) $$) - **Range**: $$ (-3, \infty) $$ - **$$ h(x) $$** - **Domain**: All real numbers ($$ (-\infty, \infty) $$) - **Range**: $$ (0, \frac{1}{27}) $$ - **$$ k(x) $$** - **Domain**: All real numbers ($$ (-\infty, \infty) $$) - **Range**: $$ (2, \infty) $$ 4. **Asymptotes:** - All functions $$ f, g, h, k $$ have a horizontal asymptote at $$ y = 0 $$. **Conclusion:** The function $$ f(x) = \left(\frac{1}{3}ight)^{x} $$ is an exponential function with a domain of all real numbers and a range of positive real numbers. When transformed, the functions $$ g, h, $$ and $$ k $$ maintain the exponential nature but shift vertically and horizontally, affecting their ranges and positions on the graph. All transformed functions share a common horizontal asymptote at $$ y = 0 $$.

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