ACT Math · Function notation and transformations

If f(x) = x² + 1 and g(x) = 2x - 3, what is (f - g)(4)?

If f(x)=x2+1f(x) = x^2 + 1 and g(x)=2x−3g(x) = 2x - 3, what is (f−g)(4)(f - g)(4)?

Answer and explanation

Answer: B: 12

The notation (f−g)(4)(f-g)(4) means f(4)−g(4)f(4)-g(4). Compute f(4)=42+1=17f(4)=4^2+1=17 and g(4)=2(4)−3=5g(4)=2(4)-3=5. Then 17−5=1217-5=12, choice B. Choice C (1717) is f(4)f(4) alone, with g(4)g(4) never subtracted. Choice A (1010) comes from using g(4)=7g(4)=7 by adding 33 instead of subtracting it. Choice D (2222) adds the two functions, 17+517+5, so it computes (f+g)(4)(f+g)(4) rather than (f−g)(4)(f-g)(4). You can also form the difference first: (f−g)(x)=x2+1−(2x−3)=x2−2x+4(f-g)(x)=x^2+1-(2x-3)=x^2-2x+4, and at x=4x=4 that equals 16−8+4=1216-8+4=12. Be careful to distribute the minus sign across all of g(x)g(x).

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