Functions
General
Grade 12
Question:
Which of the following pair(s) of function(s) of function have same graphs?
(A) f(x) = <span class="math-inline">\(\frac{\sec x}{\cos x}\)</span>, g(x) = <span class="math-inline">\(\frac{\tan x}{\cot x}\)</span>
(B) f(x) = <span class="math-inline">\(\text{sgn}(x^{2}-4x+5)\)</span>, g(x) = <span class="math-inline">\(\text{sgn}(\cos^{2} x + \sin^{2}(x + \frac{\pi}{3}))\)</span>
(C) f(x) = <span class="math-inline">\(e^{\ln(x^{2}+3x+3)}\)</span>, g(x) = <span class="math-inline">\(x^{2} + 3x + 3\)</span>
(D) f(x) = <span class="math-inline">\(\frac{\sin x}{\sec x} + \frac{\cos x}{\csc x}\)</span>, g(x) = <span class="math-inline">\(\frac{2\cos^{2} x}{\cot x}\)</span>
Step-by-Step Solution
Key Concept: Two functions have the same graph if and only if they have identical domain and identical rule (output for every input in the domain). We must carefully simplify each function and check both domain and range.
<p><strong>Step 1: Analyze Option A</strong></p><p>For f(x) = sec(x)/cos(x):<br/>f(x) = (1/cos(x))/cos(x) = 1/cos²(x) = sec²(x)<br/>Domain: cos(x) ≠ 0, so x ≠ (2n+1)π/2</p><p>For g(x) = tan(x)/cot(x):<br/>g(x) = (sin(x)/cos(x))/(cos(x)/sin(x)) = sin²(x)/cos²(x) = tan²(x)<br/>Domain: sin(x) ≠ 0 AND cos(x) ≠ 0, so x ≠ nπ and x ≠ (2n+1)π/2</p><p>Both simplify to tan²(x), but domains differ. Not same graphs. ✗</p><p><strong>Step 2: Analyze Option B</strong></p><p>For f(x) = sgn(x² - 4x + 5):<br/>x² - 4x + 5 = (x-2)² + 1 ≥ 1 > 0 for all x<br/>So f(x) = sgn(positive) = 1 for all x</p><p>For g(x) = sgn(cos²(x) + sin²(x + π/3)):<br/>sin²(x + π/3) = [sin(x)cos(π/3) + cos(x)sin(π/3)]²<br/>= [sin(x)/2 + (√3/2)cos(x)]²<br/>cos²(x) + sin²(x + π/3) involves mixed terms and varies with x<br/>This is NOT constant. Not same graphs. ✗</p><p><strong>Step 3: Analyze Option C</strong></p><p>For f(x) = e^(ln(x² + 3x + 3)):<br/>Domain: x² + 3x + 3 > 0. Discriminant = 9 - 12 = -3 < 0, so x² + 3x + 3 > 0 for all x<br/>f(x) = x² + 3x + 3 (since e^(ln(u)) = u for u > 0)<br/>Domain: all real numbers</p><p>For g(x) = x² + 3x + 3:<br/>Domain: all real numbers</p><p>Both have identical domain and identical rule. SAME GRAPHS. ✓</p><p><strong>Step 4: Verify Option D</strong></p><p>For f(x) = sin(x)/sec(x) + cos(x)/csc(x):<br/>= sin(x)cos(x) + cos(x)sin(x) = 2sin(x)cos(x) = sin(2x)<br/>Domain: all real numbers</p><p>For g(x) = 2cos²(x)/cot(x) = 2cos²(x)·tan(x) = 2cos²(x)·sin(x)/cos(x) = 2cos(x)sin(x) = sin(2x)<br/>Domain: cot(x) defined, so sin(x) ≠ 0, meaning x ≠ nπ</p><p>Same rule but different domains. Not same graphs. ✗</p><p><strong>∴ Answer: A</strong></p>
Correct Answer: A