(a) Graph $$ f(x)=2^{x}-4 $$. (b) Find the zero of $$ f(x) $$. (c) Based on the graph, solve $$ f(x)

Question

(a) Graph $$ f(x)=2^{x}-4 $$. (b) Find the zero of $$ f(x) $$. (c) Based on the graph, solve $$ f(x)<0 $$.

UpStudy AI · Accepted Answer

Sure, let's tackle each part of the problem step by step.

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### **(a) Graph of $$ f(x) = 2^{x} - 4 $$**

To graph the function $$ f(x) = 2^{x} - 4 $$, we'll analyze its key features:

1. **Base Function**: The base exponential function is $$ 2^{x} $$, which has the following characteristics:
   - **Y-intercept**: $$ (0, 1) $$ because $$ 2^{0} = 1 $$.
   - **Horizontal Asymptote**: $$ y = 0 $$ as $$ x \to -\infty $$.
   - **Growth**: Increases rapidly as $$ x $$ increases.

2. **Transformation**: The function $$ 2^{x} - 4 $$ shifts the base function **downward** by 4 units.
   - **New Y-intercept**: $$ (0, 1 - 4) = (0, -3) $$.
   - **Horizontal Asymptote**: $$ y = -4 $$.

3. **Plotting Points**:
   
   | $$ x $$ | $$ 2^{x} $$ | $$ f(x) = 2^{x} - 4 $$ |
   |---------|------------|------------------------|
   | -2      | 0.25       | -3.75                  |
   | -1      | 0.5        | -3.5                   |
   | 0       | 1          | -3                     |
   | 1       | 2          | -2                     |
   | 2       | 4          | 0                      |
   | 3       | 8          | 4                      |
   | 4       | 16         | 12                     |

4. **Graph Description**:
   - Starts near the horizontal asymptote $$ y = -4 $$ for large negative $$ x $$ values.
   - Passes through the y-intercept at $$ (0, -3) $$.
   - Crosses the x-axis at $$ x = 2 $$ (zero of the function).
   - Increases rapidly as $$ x $$ becomes positive.

*Since this is a text-based medium, please sketch the graph using the points above, ensuring the exponential growth and downward shift are represented.*

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### **(b) Finding the Zero of $$ f(x) $$**

To find the zero of the function $$ f(x) = 2^{x} - 4 $$, set $$ f(x) $$ equal to zero and solve for $$ x $$:

$$
2^{x} - 4 = 0 \
2^{x} = 4
$$

Since $$ 4 $$ is a power of $$ 2 $$, we can write:

$$
2^{x} = 2^{2} \
x = 2
$$

**Zero of $$ f(x) $$:** $$ x = 2 $$

*This means the graph of $$ f(x) = 2^{x} - 4 $$ crosses the x-axis at the point $$ (2, 0) $$.*

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### **(c) Solving $$ f(x) < 0 $$ Based on the Graph**

We need to determine the values of $$ x $$ for which $$ f(x) = 2^{x} - 4 $$ is less than zero:

$$
2^{x} - 4 < 0 \
2^{x} < 4
$$

Again, recognizing that $$ 4 = 2^{2} $$:

$$
2^{x} < 2^{2}
$$

Since the base $$ 2 $$ is greater than 1, the inequality holds when the exponent is less than 2:

$$
x < 2
$$

**Solution:** All real numbers $$ x $$ such that $$ x < 2 $$.

*Graphically, this corresponds to the portion of the graph below the x-axis, which occurs to the left of the vertical line $$ x = 2 $$.*

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### **Summary**

- **(a)** The graph of $$ f(x) = 2^{x} - 4 $$ is an exponential curve shifted downward by 4 units, with a y-intercept at $$ (0, -3) $$, a horizontal asymptote at $$ y = -4 $$, and passing through $$ (2, 0) $$.
- **(b)** The zero of $$ f(x) $$ is at $$ x = 2 $$.
- **(c)** $$ f(x) < 0 $$ for all real numbers $$ x $$ where $$ x < 2 $$.
**(a) The graph of $$ f(x) = 2^{x} - 4 $$ is an exponential curve shifted downward by 4 units, passing through the point $$ (2, 0) $$. **(b) The zero of $$ f(x) $$ is at $$ x = 2 $$. **(c) $$ f(x) < 0 $$ for all real numbers $$ x $$ where $$ x < 2 $$.

(a) Graph .
(b) Find the zero of .
© Based on the graph, solve .

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