91Ó°ÊÓ

An exponential function is one of the fundamental types of mathematical functions. In our exercise, the function is given by \[ f(x) = 10^x \] Exponential functions are known for their characteristic curve, which lies above the x-axis and increases rapidly. The graph of \( f(x) = 10^x \) passes through the point \( (0, 1) \), which is a key feature of exponential curves because any number (other than zero) raised to the power of zero is 1.

• The graph has a horizontal asymptote at \( y = 0 \), which means as x approaches negative infinity, the graph approaches the x-axis but never quite touches it.
• The domain of the function is all real numbers \( (-\infty, \infty) \).
• The range of \( f(x) \) is \( (0, \infty) \), meaning that as \( x \) moves from negative to positive values, \( f(x) \) starts from close to zero and grows towards infinity.

A horizontal compression involves adjusting the x-values to squeeze the graph of the function closer together along the horizontal axis. In our transformation, the function \[ g(x) = 10^{\frac{x+1}{2}} \] incorporates a horizontal compression alongside a horizontal shift. The algebraic manipulation of \( \frac{x+1}{2} \) causes these transformations.

• First, the \( x+1 \) means each point is moved 1 unit to the left.
• Second, dividing by 2 results in a compression by a factor of 2. This increases the speed with which \( g(x) \) reaches its maximum values compared to \( f(x) \).

Tracking a point like \( (0, 1) \), this transformation shifts it to \( (-1, 10^{(\frac{1}{2})}) = (-1, \sqrt{10}) \). This point reflects the compressed nature of \( g(x) \).

Vertical shifts involve moving the graph up or down along the y-axis without altering its shape. For \[ g(x) = 10^{\frac{x+1}{2}} - 20 \] there is a vertical shift downward by 20 units as shown by the \( -20 \) in the equation.

• This shift moves every point down by 20. Therefore, a point like \( (0, \sqrt{10}) \) moves to \( (0, \sqrt{10} - 20) \).
• More importantly, the horizontal asymptote of \( y = 0 \) shifts to \( y = -20 \).

A vertical shift does not affect the domain but does change the range significantly. The function now appears lower on the graph but retains its exponential rise.

The domain and range of a function describe the set of possible input (x-values) and output (y-values) respectively. For our original function \[ f(x) = 10^x \], the domain is all real numbers \( (-\infty, \infty) \), indicating that you can input any real number into the function. The range is \( (0, \infty) \), meaning the outputs are always positive. For the transformed function \[ g(x) = 10^{\frac{x+1}{2}} - 20 \], the domain remains \( (-\infty, \infty) \). This is because transformations like shifts and compressions don't limit or expand the input possibilities. However, the range changes. Since we have a vertical shift downward by 20:

• The new range of \( g(x) \) is \( (-20, \infty) \).
• This implies \( g(x) \) can still reach any value above \(-20\), but never reach or drop below \( -20 \).

Understanding the domain and range in these transformations helps ensure we know the limits and extent of the function's behavior.