📝 Independent and dependent variables in functions (12 MCQs)

📖 From Calculus • 1. Basics before calculus • 12 questions available

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What is Independent and dependent variables in functions?

Definition:
The independent variable is the input of a function, typically denoted as $x$ , which can be freely chosen, while the dependent variable is the output, denoted as $y$ or $f(x)$ , whose value depends directly on the chosen input value.

Example:
In $y = 3x^2$ , if $x = 2$ (independent), then $y = 3(2)^2 = 12$ (dependent), so $y$ changes with $x$ .

Reason:
This distinction helps in understanding cause-and-effect relationships, enabling us to control inputs to observe how outputs respond, which is crucial in experiments and applied sciences.

Easy

Medium

Hard

📝 All Independent and dependent variables in functions MCQs

Q1. In the expression $y = f(x)$ , which variable is the independent variable?

x

✅

y

C.f

D.Both

x

and

y

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The independent variable is the one you are free to choose; it appears inside the function parentheses. In $y = f(x)$ , $x$ is the argument you can vary, and the function then determines a corresponding $y$ . Therefore $x$ is the independent variable.

Q2. Given the pairs (0,3), (1,4), (2,-1), (3,6) that define a function f, which statement must always hold for any function containing these pairs?

A.Two different x-values cannot produce the same y-value.

B.A single x-value cannot correspond to two different y-values. ✅

C.The function must be linear.

D.The function must be increasing.

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: By definition, a function assigns exactly one output to each input. Therefore, the same x cannot map to two distinct y-values. The other statements are not required; a function may be non‑linear or decreasing, and different x-values may share the same y.

Q3. In experiment 1 (plant height vs. sunlight hours) and experiment 2 (distance vs. time), which pair correctly identifies the independent variables?

A.Plant height and distance

B.Sunlight hours and distance

C.Plant height and time

D.Sunlight hours and time ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: The independent variable is the one that the researcher controls or varies. In the first experiment the researcher varies sunlight hours, and in the second experiment time is varied. Thus the correct pair of independent variables is sunlight hours together with time.

Q4. A biologist varies the amount of nutrient solution (x) and records the resulting bacterial count (y). Which variable is dependent?

A.Amount of nutrient solution

B.Bacterial count ✅

C.Both variables

D.Neither variable

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The dependent variable is the outcome that changes in response to the independent variable. Here the biologist changes the nutrient amount (x) and observes how the bacterial count (y) responds, making the bacterial count the dependent variable.

Q5. For the linear function $y = 2x + 5$ , if the independent variable x increases by 3 units, how does the dependent variable y change?

A.Increases by 3

B.Increases by 2

C.Increases by 5

D.Increases by 6 ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: In a linear function the slope indicates the change in y for each unit change in x. The slope here is 2, so an increase of 3 in x produces a change of $2 \times 3 = 6$ in y. Hence y increases by 6.

Q6. Given $f(x) = x^{2}$ and $g(x) = 2x$ , which function has the larger instantaneous rate of change at $x = 3$ ?

A.g(x)

B.Both have the same rate

C.Neither can be determined

D.f(x) ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The instantaneous rate of change is given by the derivative. For $f(x) = x^{2}$ , $f'(x) = 2x$ , which at $x = 3$ equals 6. For $g(x) = 2x$ , $g'(x) = 2$ . Since 6 > 2, the function $f(x)$ has the larger rate of change.

Q7. In the equation $s = vt$ where speed v is constant, which variable directly depends on time t?

A.Speed v

B.Displacement s ✅

C.Both s and v

D.Neither s nor v

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: When speed is constant, displacement is obtained by multiplying speed by time. Thus as time changes, displacement changes proportionally, making s the variable that directly depends on t. Speed v remains unchanged because it is held constant.

Q8. A function f satisfies f(1)=4 and f(2)=7. Assuming f is linear, what is f(3)?

A.11

B.12

C.10 ✅

D.13

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: A linear function has constant slope. The slope between the given points is $(7-4)/(2-1) = 3$ . Adding this slope to f(2) gives $7 + 3 = 10$ . Therefore f(3) equals 10, which corresponds to option C.

Q9. Compare the role of the independent variable in the linear function $y = 3x + 2$ and the exponential function $y = 5^{x}$ . Which statement is accurate?

A.Both functions have the same rate of change with respect to x.

B.In both functions, changing x changes y, but the relationship is additive in the linear case and multiplicative in the exponential case. ✅

C.In the linear function x is dependent, while in the exponential function x is independent.

D.Only the linear function has an independent variable.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The independent variable x influences y in both cases, but the manner differs. In the linear function the effect of x adds a constant amount per unit change, while in the exponential function each unit change multiplies y by a constant factor. Hence the additive versus multiplicative description is correct.

Q10. Let $g(x) = 3x + 1$ and $f(u) = u^{2} - 4$ . Define the composite $h(x) = f(g(x))$ . Which variable serves as the independent variable for $h$ ?

A.g(x)

B.x ✅

C.u

D.Both x and u

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: In the composite function, the outer function f receives the output of g as its input. However, the original argument that can be varied is x. Changing x changes g(x) and consequently h(x). Therefore x remains the independent variable for the composite function.

Q11. Suppose $f(x) = 2x + 1$ and $g(x) = x^{2}$ . Define $R(x) = \frac{f(x)}{g(x)}$ . If x doubles, how does the ratio R change?

A.It more than doubles

B.It exactly doubles

C.It less than doubles ✅

D.It remains unchanged

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: R(x) = (2x+1)/x^{2}. Replacing x with 2x gives R(2x) = (4x+1)/(4x^{2}). Comparing R(2x) to R(x) shows the numerator grows linearly while the denominator grows quadratically, causing the overall ratio to decrease. Hence the ratio increases by less than a factor of two.

Q12. A study records three variables: dosage (d), response time (t), and effect magnitude (e). The researchers state that e depends on both d and t, while d is set by the experimenter and t is measured after dosage is administered. Which variable(s) are independent?

A.Only dosage d

B.Only response time t

C.Both dosage d and response time t ✅

D.Effect magnitude e

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Independence refers to variables that are not determined by other variables in the model. Dosage is chosen by the experimenter, making it independent. Response time, although measured after dosage, is not

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