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Zorluk: ZorFunctions and Custom Symbol Operations

For all real numbers xx, the function ff is defined by f(x)=x2−4x+7f(x) = x^2 - 4x + 7 and the function gg is defined by g(x)=2x+1g(x) = 2x + 1. If f(g(k))=19f(g(k)) = 19 and k<0k < 0, what is the value of kk?

Cevap: -1.5

Cevap

The value of kk is −1.5-1.5.
To solve for kk, substitute g(k)=2k+1g(k) = 2k + 1 into f(x)f(x) to get f(g(k))=(2k+1)2−4(2k+1)+7f(g(k)) = (2k + 1)^2 - 4(2k + 1) + 7. Expanding and combining like terms yields 4k2−4k+44k^2 - 4k + 4. Setting this equal to 1919 gives 4k2−4k−15=04k^2 - 4k - 15 = 0. Factoring the quadratic expression yields (2k−5)(2k+3)=0(2k - 5)(2k + 3) = 0, which gives k=52=2.5k = \frac{5}{2} = 2.5 or k=−32=−1.5k = -\frac{3}{2} = -1.5. Since the problem specifies that k<0k < 0, the value of kk must be −1.5-1.5.

Adım Adım Çözüm

1
Express the nested composite function f(g(k))f(g(k)) in terms of kk
f(g(k))=(2k+1)2−4(2k+1)+7f(g(k)) = (2k + 1)^2 - 4(2k + 1) + 7
Substituting g(k)=2k+1g(k) = 2k + 1 for every occurrence of xx in f(x)f(x) sets up the composite function evaluation.
2
Expand terms and simplify the algebraic expression
f(g(k))=4k2−4k+4f(g(k)) = 4k^2 - 4k + 4
Expanding (2k+1)2=4k2+4k+1(2k + 1)^2 = 4k^2 + 4k + 1 and distributing −4(2k+1)=−8k−4-4(2k + 1) = -8k - 4, then combining like terms yields 4k2−4k+44k^2 - 4k + 4.
3
Set the composite function expression equal to 19 and form a standard quadratic equation
4k2−4k−15=04k^2 - 4k - 15 = 0
Subtracting 1919 from both sides produces a standard quadratic equation ax2+bx+c=0ax^2 + bx + c = 0.
4
Solve the quadratic equation and apply the condition k<0k < 0
k=−1.5k = -1.5
Factoring (2k−5)(2k+3)=0(2k - 5)(2k + 3) = 0 gives k=2.5k = 2.5 or k=−1.5k = -1.5. The constraint k<0k < 0 selects k=−1.5k = -1.5.

Anahtar Kavram

Composite function evaluation f(g(x))f(g(x)) combined with quadratic equation solving under domain constraints.
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Functions and Custom Symbol Operations

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