b) Given the function f(t) = 1 − 3/t and the range −1, 3, 7 we need to find the domain.

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ScanSolve AI Tutor · Accepted Answer

Here are the solutions to your problems. b) Given the function f(t) = 1 − 3/t and the range −1, 3, 7, we need to find the domain. The range consists of the output values of f(t). We set f(t) equal to each value in the range and solve for t. Step 1: Set f(t) = y and solve for t in terms of y. y = 1 − 3/t; y − 1 = −3/t; 3/t = 1 − y; t = (3)/(1 − y) Step 2: Substitute each value from the range into the expression for t. For y = −1: t = (3)/(1 − (−1)) = (3)/(1 + 1) = 3/2; For y = 3: t = (3)/(1 − 3) = (3)/(−2) = −3/2; For y = 7: t = (3)/(1 − 7) = (3)/(−6) = −1/2 The domain is the set of these t values. The domain is −3/2, −1/2, 3/2. a) Given a^(x−3)b^(3x) = a^(5−x)b^(5x), we need to show that x log(a/b) = 4 log a. Step 1: Group terms with the same base on opposite sides of the equation. (a^(x−3))/(a^(5−x)) = (b^(5x))/(b^(3x)); Step 2: Apply the exponent rule (m^p)/(m^q) = m^(p−q). a^((x−3) − (5−x)) = b^(5x − 3x); a^(x−3−5+x) = b^(2x); a^(2x−8) = b^(2x) Step 3: Take the logarithm of both sides. log(a^(2x−8)) = log(b^(2x)); Step 4: Apply the logarithm rule log(M^p) = p log M. (2x−8)log a = 2x log b Step 5: Distribute and rearrange the terms to isolate x. 2x log a − 8 log a = 2x log b; 2x log a − 2x log b = 8 log a Step 6: Factor out 2x from the left side. 2x (log a − log b) = 8 log a; Step 7: Apply the logarithm rule log M − log N = log(M/N). 2x log(a/b) = 8 log a Step 8: Divide both sides by 2. x log(a/b) = 4 log a This completes the proof.

b) Given the function f(t) = 1 − (3)/(t) and the range \-1, 3, 7 we need to find the domain.

Answer

Step-by-step solution

Part b· 2 steps

Set $f(t) = y$ and solve for $t$ in terms of $y$ .

Substitute each value from the range into the expression for $t$ .

Part a· 8 steps

Group terms with the same base on opposite sides of the equation.

Apply the exponent rule $\frac{m^p}{m^q} = m^{p-q}$ .

Take the logarithm of both sides.

Apply the logarithm rule $\log(M^p) = p \log M$ .

Distribute and rearrange the terms to isolate $x$ .

Factor out $2x$ from the left side.

Apply the logarithm rule $\log M - \log N = \log\left(\frac{M}{N}\right)$ .

Divide both sides by 2.

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