$q(x)=-4 \cdot 10^x$

The function q ( x ) = − 4 ⋅ 1 0 x is an exponential function with a domain of ( − ∞ , ∞ ) and a range of ( − ∞ , 0 ) . Its y-intercept is at ( 0 , − 4 ) , there is no x-intercept, and it decreases for all values of x with a horizontal asymptote at y = 0 .
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Answered by Anonymous | 2025-07-04

The function q ( x ) = − 4 \t × 1 0 x is an exponential function.
The domain is ( − ∞ , ∞ ) and the range is ( − ∞ , 0 ) .
The y-intercept is ( 0 , − 4 ) , and there is no x-intercept.
The function is decreasing with a horizontal asymptote at y = 0 . q ( x ) = − 4 \t × 1 0 x ​

Explanation

Understanding the Function We are given the function q ( x ) = − 4 \t × 1 0 x . Our goal is to analyze its properties, including its type, domain, range, intercepts, asymptotic behavior, and whether it's increasing or decreasing, and then sketch its graph.

Identifying the Function Type The function q ( x ) = − 4 \t × 1 0 x is an exponential function. The base of the exponent is 10.

Determining the Domain The domain of an exponential function is all real numbers. Therefore, the domain of q ( x ) is ( − ∞ , ∞ ) .

Determining the Range Since 1 0 x is always positive, − 4 \t × 1 0 x is always negative. As x approaches − ∞ , 1 0 x approaches 0, so q ( x ) approaches 0. As x approaches ∞ , 1 0 x approaches ∞ , so q ( x ) approaches − ∞ . Therefore, the range of q ( x ) is ( − ∞ , 0 ) .

Finding Intercepts To find the y-intercept, we set x = 0 : q ( 0 ) = − 4 \t × 1 0 0 = − 4 \t × 1 = − 4 . So the y-intercept is ( 0 , − 4 ) .
To find the x-intercept, we set q ( x ) = 0 : − 4 \t × 1 0 x = 0 . However, 1 0 x is never 0, so there is no x-intercept.

Analyzing Asymptotic Behavior As x approaches − ∞ , q ( x ) approaches 0. Thus, y = 0 is a horizontal asymptote. As x approaches ∞ , q ( x ) approaches − ∞ .

Determining Increasing or Decreasing To determine if the function is increasing or decreasing, we can analyze its derivative. The derivative of q ( x ) = − 4 \t × 1 0 x is q ′ ( x ) = − 4 \t × ln ( 10 ) \t × 1 0 x . Since 1 0 x is always positive and ln ( 10 ) is positive, q ′ ( x ) is always negative. Therefore, the function is decreasing for all x .

Summary of Function Properties In summary, the function q ( x ) = − 4 \t × 1 0 x is an exponential function with a domain of ( − ∞ , ∞ ) , a range of ( − ∞ , 0 ) , a y-intercept at ( 0 , − 4 ) , no x-intercept, a horizontal asymptote at y = 0 , and is decreasing for all x .


Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in a bank account that earns compound interest, the amount of money you have will grow exponentially over time. Similarly, the decay of a radioactive substance can be modeled using an exponential function.

Answered by GinnyAnswer | 2025-07-04