activity - comparing non-linear functions

activity
value: 1

which function intersects the x-axis?

a. (h(x) = x^2 - 5)
b. (g(x) = (x + 3)^2 + 2)
c. (p(x) = -9)
d. (m(x) = 4x^2 + 5)

Question

activity - comparing non-linear functions

activity
value: 1

which function intersects the x-axis?

a. (h(x) = x^2 - 5)
b. (g(x) = (x + 3)^2 + 2)
c. (p(x) = -9)
d. (m(x) = 4x^2 + 5)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "x-intercept", "Quadratic Equations" ], "new_concepts": [], "current_concepts": [ "x-intercept", "Quadratic Equations", "Domain of Functions" ] } ## Define the condition for x-axis intersection $$ f(x) = 0 $$ ## Test each function for real solutions $$ \begin{aligned} &\text{For } a:\, x^2 - 5 = 0 \implies x^2 = 5 \implies x = \pm\sqrt{5} \quad (\text{Real roots})\ &\text{For } b:\, (x + 3)^2 + 2 = 0 \implies (x + 3)^2 = -2 \quad (\text{No real roots})\ &\text{For } c:\, p(x) = -9 \implies -9 = 0 \quad (\text{No solution})\ &\text{For } d:\, 4x^2 + 5 = 0 \implies 4x^2 = -5 \implies x^2 = -\frac{5}{4} \quad (\text{No real roots}) \end{aligned} $$ ## Identify the correct option $$ h(x) = x^2 - 5 \text{ intersects the x-axis at } (\sqrt{5}, 0) \text{ and } (-\sqrt{5}, 0). $$

a. $h(x) = x^2 - 5$

b. $g(x) = (x + 3)^2 + 2$

c. $p(x) = -9$

d. $m(x) = 4x^2 + 5$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "x-intercept" ] }

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QUESTION IMAGE

Question

Explanation:

Define the condition for x-axis intersection

Test each function for real solutions

Identify the correct option

Answer: