Problem 40 - Algebra and Trigonometry

Question

The figure shows the graph of \(f(x)=e^{x}\), use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing urility to confirm your hand-drawn graphs. $$ h(x)=e^{x+1}-1 $$

Step-by-Step Solution

Answer

The graph of \(h(x)=e^{x+1}-1\) is the graph of \(f(x)=e^{x}\) shifted to the left by 1 unit and down by 1 unit. The asymptote of \(h(x)=e^{x+1}-1\) is at y = -1. The domain of h(x) is \(-\infty < x < \infty\) and the range is \(-1 < y < \infty\).

Step 1: Identify the transformations

The function \(h(x)=e^{x+1}-1\) is the base function \(f(x)=e^{x}\) shifted to the left by 1 unit and down by 1 unit. The +1 inside the exponent 'x+1' corresponds to shift to the left by 1 and '-1' outside the exponent represents a shift down by 1 unit.

Step 2: Calculate the asymptote

An asymptote in this case is a line that the graph approaches but never touches. In \(f(x)=e^{x}\) the x-axis i.e. y=0 is the horizontal asymptote. As our function is a shift of the base function, the horizontal asymptote of \(h(x)=e^{x+1}-1\) will be y = -1.

Step 3: Determine Domain and Range

The domain of a function is the complete set of possible values of the independent variable, in this case 'x'. Since there is no restriction on the values x can assume, therefore, the domain of this function like the base function will be all real numbers or \(-\infty < x < \infty\). The range of a function is the complete set of possible values of the dependent variable, in this case 'y'. In the base function, y can take any value greater than 0 but in our function, since it is shifted down by 1 unit, y will be greater than -1 or \(-1 < y < \infty\).

Step 4: Graphing the function

To graph \(h(x)=e^{x+1}-1\), start by drawing the graph of \(f(x)=e^{x}\) then shift it one unit to the left and one unit down. Confirm your hand-drawn graph with a graphing utility.

Key Concepts

Asymptotes

An asymptote is a line that a graph approaches but never quite touches. It acts like a boundary for the graph. In exponential functions, horizontal asymptotes are common. For the base function \(f(x) = e^x\), the horizontal asymptote is the line \(y = 0\), or the x-axis.
When transforming a function, the asymptote can shift. With the function \(h(x) = e^{x+1} - 1\), the graph of \(f(x) = e^x\) is shifted down by 1 unit. As a result, the horizontal asymptote becomes \(y = -1\). This means that no matter how far the graph stretches, it approaches \(y = -1\) but never reaches it.
Understanding asymptotes is important as they help predict the behavior of graphs, especially far from the origin or in extreme values.

Domain and Range

The domain of a function comprises all the possible x-values that can be input into the function. For exponential functions like \(f(x) = e^x\), the domain is all real numbers because you can exponentiate \(e\) with any real number.
When you perform transformations, the domain often remains unchanged unless there's a horizontal compression/stretch or reflection. Thus, for the function \(h(x) = e^{x+1} - 1\), the domain remains \(-\infty, \infty\) because the plus one in the exponent only shifts the graph horizontally.
The range of a function is about the possible y-values. For \(f(x) = e^x\), the range is \(y > 0\) since \(e^x\) is always positive. After shifting the graph down by 1 unit for \(h(x)\), the smallest y-value becomes just over \(-1\). Thus, the range is \-1 < y < \infty\.

Exponential Function

Exponential functions are a special type of mathematical function where a constant base is raised to a variable exponent, like \(f(x) = e^x\). They have unique properties:

• They grow rapidly as \(x\) increases.
• They have a horizontal asymptote, typically at \(y=0\).
• The base, \(e\), is an irrational number approximately equal to 2.71828.

Understanding these characteristics is crucial, as they define the function's swift increase and its approach toward but never reaching the x-axis. This rapid growth and unique properties make exponential functions applicable in various real-world scenarios, such as population growth or radioactive decay.

Function Transformation

Function transformation involves modifying a function to change its graph's position or shape. This includes shifting, reflecting, stretching, or compressing.
For \(h(x)=e^{x+1}-1\), the transformation involves shifting the base function \(f(x)=e^x\). The \(+1\) inside the exponent indicates a leftward horizontal shift by 1 unit, while the \(-1\) outside the exponent represents a vertical shift downward by 1 unit.

• Horizontal shifts affect the graph's position left or right.
• Vertical shifts move it up or down.
• Reflections would flip it over an axis.
• Stretching or compressing changes its scaling.

Understanding these transformations helps in sketching the function's graph more accurately and understanding its behavior in different contexts.