Evaluate each function below. 52. If f(x) = 2x + 7, find f(8). 54. Find the range of the function f(x) = x^2 - 6x if the domain is { -7, -2, 1}. 55. Given the graph of f(x) below,... Evaluate each function below. 52. If f(x) = 2x + 7, find f(8). 54. Find the range of the function f(x) = x^2 - 6x if the domain is { -7, -2, 1}. 55. Given the graph of f(x) below, find f(2).

Understand the Problem
The question is asking about relations and functions in mathematics, specifically how to determine the domain and range of given functions and evaluate specific function values.
Answer
For problem 52, \( f(8) = 23 \). For problem 54, range is \( R = \{91, 16, -5\} \).
Answer for screen readers
For problem 52, ( f(8) = 23 ).
For problem 54, the range is ( R = {91, 16, -5} ).
Steps to Solve
- Determine Domain and Range from Relations
For the sets of coordinates in each table or graph:
- Domain (D): List all unique x-values.
- Range (R): List all unique y-values.
For example, for coordinates (((2, 2), (-1, -1), (0, -1), (4, 2))):
- Domain: ( D = {2, -1, 0, 4} )
- Range: ( R = {2, -1} )
- Check if Relation is a Function
A relation is a function if each input (x-value) corresponds to exactly one output (y-value).
- In the first set, there are unique x-values, so it is a function.
- In the second set, if any x-value repeats with a different y-value, it's not a function.
- Evaluate the Function
For ( f(x) = 2x + 7 ):
-
Substitute ( x = 8 ):
$$ f(8) = 2(8) + 7 = 16 + 7 = 23 $$
- Finding the Range for a Function
Given ( f(x) = x^2 - 6x ) with the domain ( { -7, -2, 1 } ):
-
Calculate ( f(x) ) for each x-value:
$$ f(-7) = (-7)^2 - 6(-7) = 49 + 42 = 91 $$
$$ f(-2) = (-2)^2 - 6(-2) = 4 + 12 = 16 $$
$$ f(1) = (1)^2 - 6(1) = 1 - 6 = -5 $$
Range is ( R = {91, 16, -5} ).
- Finding the Value from the Graph
For the function based on the graph, to find ( f(2) ), locate where ( x = 2 ) on the graph and read the corresponding y-value.
For problem 52, ( f(8) = 23 ).
For problem 54, the range is ( R = {91, 16, -5} ).
More Information
- A function maps each input to exactly one output.
- The domain represents all possible x-values, while the range represents all possible y-values.
- Evaluating a function involves substituting specific x-values into the function expression.
Tips
- Misidentifying a relation as a function when x-values repeat with different y-values.
- Forgetting to list unique values for the domain and range.
- Incorrectly calculating function values due to misapplication of mathematical operations.
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