Question

Which statements about the graph of the exponential function f(x) are TRUE?The x-intercept is 1.The y-intercept is 3.The asymptote is y = -3The range is all real numbers greater than -3The domain is all real numbers.f(x) is positive for all x-values greater than 1As x increases, f(x) approaches, but never reaches, -3.

Answer 1

The following statements about the graph of the exponential function f(x) are TRUE:

The asymptote is y = -3.

The range is all real numbers greater than -3.

The domain is all real numbers.

As x increases, f(x) approaches, but never reaches, -3.

The other statements are FALSE:

The x-intercept is 1.

The y-intercept is 3.

f(x) is positive for all x-values greater than 1.

The x-intercept is the point where the graph of the function crosses the x-axis. The graph of an exponential function never crosses the x-axis, so the x-intercept is undefined.

The y-intercept is the point where the graph of the function crosses the y-axis. The y-intercept of an exponential function is always (0, 1).

The range of a function is the set of all possible output values. The graph of an exponential function approaches, but never reaches, its horizontal asymptote. Therefore, the range of an exponential function is the set of all real numbers greater than or equal to its vertical asymptote.

The domain of a function is the set of all possible input values. Exponential functions are defined for all real numbers.

As x increases, f(x) approaches, but never reaches, its horizontal asymptote. This is the most important property of exponential functions.

The image that you have provided shows the graph of an exponential function with a horizontal asymptote at y = -3. The graph approaches, but never reaches, the asymptote as x increases.

Therefore, the only statements that are TRUE about the graph of the exponential function f(x) are the following:

The asymptote is y = -3.

The range is all real numbers greater than -3.

The domain is all real numbers.

As x increases, f(x) approaches, but never reaches, -3.

Answer 2

1 The x-intercept is the value of x where the graph intersects the x-axis. The graph crosses the x-axis at x = 1. This statement is true.

2 The y-intercept is the value of y where the graph intersects the y-axis. The graph crosses the y-axis at y = -2. This statement is false.

3 The horizontal asymptote is the value of y to which the graph approaches but never reaches. This value seems to be y = -3, thus this statement is true.

4 The range is the set of values of y where the function exists. The graph exists only for values of y greater than -3. This statement is true.

5 We can give x any real value and the function exists, i.e., any vertical line would eventually intersect the graph. This statement is true.

To find the domain of a function when we are given the graph, we use the vertical line test. This consists of drawing an imaginary vertical line throughout the x-axis. If the line intersects the graph, that value of x is part of the domain.

This imaginary exercise gives us the centainty that there is no value of x that won't intercept the graph, thus the domain is the set of all the real values.

6 We can see the graph is positive exactly when the function has its x-intercept, thus This statement is true.

7 As x increases, y goes to infinity. The value of -3 is not a number where f(x) approaches when x increases, but when x decreases. This statement is false.