Problem 18 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the graph of the function to answer the questions. (a) Determine the domain of the function. (b) Determine the range of the function. (c) Find the value(s) of \(x\) for which \(f(x)=0\). (d) What are the values of \(x\) from part (c) referred to graphically? (e) Find \(f(0)\), if possible. (f) What is the value from part (e) referred to graphically? (g) What is the value of \(f\) at \(x=1 ?\) What are the coordinates of the point? (h) What is the value of \(f\) at \(x=-1 ?\) What are the coordinates of the point? (i) The coordinates of the point on the graph of \(f\) at which \(x=-3\) can be labeled \((-3, f(-3))\), or \((-3, )\) $$f(x)=\left\\{\begin{array}{ll} x+4, & x \leq 0 \\ 4-x^{2}, & x>0 \end{array}\right.$$

Step-by-Step Solution

Answer

The domain is (-∞, ∞) and the range is (-∞, 4). The values of x for which f(x) = 0 are -4, 2 and -2. The graph intercepts the x-axis at these x-values and intercepts the y-axis at y=4. The coordinates of the points at x=1 and x=-1 are (1,3) and (-1,3) respectively, and the point for x=-3 is labelled (-3,1).

Step 1: Find the Domain

To find the domain of the function, identify the values of x that will make the function valid. Since the given function is defined for all real numbers, the domain is all real numbers or (-∞, ∞).

Step 2: Find the Range

The range of a function is the set of all possible output values (y-values). For x ≤ 0, the function is \(x+4\), which yields a minimum value at x=0, y=4. For x > 0, the function is \(4 - x^{2}\) which yields a maximum value at x=0, y=4. Therefore, the range of the function is (-∞, 4].

Step 3: Find the Value of x for which f(x) = 0

Set the function f(x) equal to zero and solve for x. For x ≤ 0, we have x+4 = 0; solving gives x = -4. For x > 0, we have 4 - x^{2} = 0; solving gives x = ±2. Therefore, the values of x for which f(x) = 0 are -4, 2 and -2.

Step 4 and 5: Graphical Interpretation

The values of x found in step 3 represent the x-intercepts of the function graph. Therefore, -4 and ±2 are the x-values where the graph of the function intercepts the x-axis. f(0) refers to the value of the function at x=0. Substituting into f(x), we get f(0) = 4, which is a point on the y-axis or y-intercept on the graph.

Step 6: Find the Value of f at x=1, and x=-1

We substitute these values into the function to find the corresponding y-values: f(1) = 4 - 1^2 = 3 and f(-1) = -1 + 4 = 3, so the coordinates of these points are (1, 3) and (-1, 3).

Step 7: Label the Point for x=-3

Lastly, the point at x=-3 would be labelled (-3, f(-3)) and upon substituting x=-3 into the function, we get f(-3) = -3 + 4 = 1, so the point is (-3, 1).

Key Concepts

Domain and Range of a Function

Understanding the domain and range of a function is fundamental in graphing and analyzing functions. The domain refers to all the possible input values, or 'x' values, that can be applied to the function without causing any mathematical errors, like dividing by zero or taking the square root of a negative number. In the given exercise, the domain is all real numbers, often expressed as \( (-\infty, \infty) \).

The range, on the other hand, describes all the possible output values, or 'y' values, that the function can produce. The range is determined by evaluating how the function behaves for all values within its domain. In this case, the range is \( (-\infty, 4] \) as the function's outputs will never exceed 4, which is the maximum y-value. It's essential to note that the domain and range provide the 'limits' within which the function lives on a graph.

X-Intercepts

The x-intercepts of a function are the points where the graph intersects the x-axis. These are the points where the output value, or 'y', is zero. To find the x-intercepts, you set the function equal to zero and solve for 'x'. In our exercise, when setting the function \( f(x) = 0 \), you get \( x = -4, 2, \text{and} -2 \), meaning these values are where the function crosses the x-axis. Identifying x-intercepts is crucial as they often represent important characteristics of the function, such as points of balance or the locations where the function changes from positive to negative output.

Y-Intercept

The y-intercept is the point where the graph of the function crosses the y-axis. To find it, you simply evaluate the function at \( x = 0 \). This point is significant as it often gives you useful information about the starting or ending point of a function in its graphical representation. In the exercise, evaluating \( f(0) \) gave us the point (0,4), which is the y-intercept. It represents the height at which the function begins or ends on the y-axis when graphed.

Function Evaluation

Evaluating a function, often referred to as function evaluation, involves finding the output value of a function for a particular input value. It's a pivotal skill in understanding how a function behaves at specific points. In the exercise, you are asked to evaluate the function at \( x = 1 \) and \( x = -1 \). Substituting these 'x' values into the function gives you the coordinates (1,3) and (-1,3), respectively. Function evaluation helps in plotting points on the graph and understanding the function's behavior at given values of 'x'.