An exponential function with range y<1,y∈R and a y-intercept of 0.

An exponential function of the form \( y = a \cdot b^x \) can have a range \( y < 1 \) if it asymptotically approaches 1 from below. Given that the y-intercept is 0, we need to consider the properties of the exponential function and make necessary adjustments to fit the conditions.

To achieve this, let's start with the general form:

\[ y = a \cdot b^x \]

For the y-intercept to be 0:
\[ a \cdot b^0 = a = 0 \]

This suggests that \( a \) must be zero, but then the function \( y = 0 \) does not have a range of \( y < 1 \). Instead, we can consider a modified exponential function such as:

\[ y = 1 - b^x \]

Here, when \( x = 0 \):
\[ y = 1 - b^0 = 1 - 1 = 0 \]

For \( y < 1 \), \( b^x \) must be positive and less than 1 for all \( x \). This means \( 0 < b < 1 \).

Therefore, an exponential function with the desired properties is:

\[ y = 1 - b^x \]

where \( 0 < b < 1 \).

### Example
For example, if \( b = 0.5 \):

\[ y = 1 - 0.5^x \]

This function has a range \( y < 1 \) and a y-intercept of 0.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How do you determine the range of an exponential function?
2. What are the properties of exponential functions?
3. How do exponential growth and decay differ?
4. What are some real-world applications of exponential functions?
5. How do transformations affect the graph of an exponential function?

**Tip:** Remember, the base of an exponential function greatly influences its growth rate. The base greater than 1 results in growth, while a base between 0 and 1 results in decay.

Explore how to construct an exponential function that satisfies the conditions y < 1 and has a y-intercept of 0. Learn about the properties of exponential decay and asymptotic behavior, with examples and detailed explanations.

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