Unit 4 of 5
Study guide for CLEP CLEP Precalculus — Unit 4: Functions & Modeling. Practice questions, key concepts, and exam tips.
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Tom is a manager at a retail store. He notices that the store's daily revenue, in dollars, can be modeled by the function f(x) = 200x + 500, where x is the number of customers. The store's daily cost, in dollars, can be modeled by the function g(x) = 50x + 200. Which of the following functions represents the daily profit?
Answer: C — The correct answer is B because profit is equal to revenue minus cost. So, the daily profit can be found by subtracting the cost function g(x) from the revenue function f(x). Option A is incorrect because it subtracts an additional 200. Option C is incorrect because it adds the cost and revenue functions. Option D is incorrect because it subtracts the revenue function from the cost function, which would give a negative profit.
Suppose we have two functions, f(x) = 2x + 1 and g(x) = x - 3. What is the result of (f ∘ g)(x), where ∘ denotes function composition?
Answer: A — The correct answer is A) 2x - 5 because (f ∘ g)(x) means applying g first and then f. So, we substitute g(x) into f: f(g(x)) = f(x - 3) = 2(x - 3) + 1 = 2x - 6 + 1 = 2x - 5. Options B, C, and D are incorrect because they do not accurately represent the composition of f and g.
Tom is a manager at a store, and he wants to analyze the revenue and cost of his business over time. He defines two functions: R(x) as the revenue and C(x) as the cost, where x is the number of months since the business started. If R(x) = 1000x + 500 and C(x) = 500x + 200, what is the profit function P(x), which is defined as the difference between the revenue and cost functions?
Answer: C — The profit function P(x) is defined as the difference between the revenue and cost functions, so P(x) = R(x) - C(x). Plugging in the given functions, we get P(x) = (1000x + 500) - (500x + 200). Simplifying this expression gives P(x) = 500x + 300. Therefore, option C is the correct answer. Options A, B, and D are incorrect because they do not represent the difference between the revenue and cost functions.
Tom is a manager at a retail store. He notices that the sales of a product can be modeled by the function f(x) = 2x + 5, where x is the number of days since the product was released and f(x) is the total sales. The cost of storing the product can be modeled by the function g(x) = x^2 + 2x + 1. What is the function that models the profit of the product, assuming the profit is the difference between the sales and the cost?
Answer: C — The correct answer is C because the profit is calculated by subtracting the cost from the sales. So, the profit function is f(x) - g(x) = (2x + 5) - (x^2 + 2x + 1) = -x^2 - 0x + 4. Option A is close but does not simplify the expression. Option B is incorrect because it adds the sales and cost functions instead of subtracting them. Option D is incorrect because it adds the sales and cost functions and also does not represent the profit.
Which of the following is a correct statement about a function?
Answer: B — Correct answer B is the definition of a function. A function is a relation between a set of inputs (the domain) and a set of possible outputs (the range). The other options are incorrect because they either swap the terms 'domain' and 'range' or use the term 'codomain', which is not the standard term used in this context. Option A is incorrect because the domain is the set of inputs and the range is the set of possible outputs. Option C is incorrect because it uses the term 'codomain', which refers to the set of all possible outputs, but in the context of a function, 'range' is the more commonly used term. Option D is incorrect because it swaps the terms 'codomain' and 'domain'.
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