Math / Algebra

QuestionThe functions $g$ and $t$ are defined for $x \in \mathbb{R}$ as follows: $\begin{array}{l} g: x \rightarrow 4x - 5 \\ t: x \rightarrow x^2 - 5x + 1 \end{array}$ (a) Find $t(6)$
(b) Show that $t(g(x)) = 16x^2 - 60x + 51$

Studdy Solution

STEP 1

1. We are given two functions: $g(x) = 4x - 5$ and $t(x) = x^2 - 5x + 1$ .
2. We need to evaluate $t(6)$ .
3. We need to show that $t(g(x)) = 16x^2 - 60x + 51$ .

STEP 2

1. Evaluate $t(6)$ .
2. Substitute $g(x)$ into $t(x)$ to find $t(g(x))$ .
3. Simplify the expression for $t(g(x))$ .
4. Verify that the simplified expression matches $16x^2 - 60x + 51$ .

STEP 3

Substitute $x = 6$ into $t(x)$ :
$t(6) = 6^2 - 5(6) + 1$

STEP 4

Simplify the expression:
$t(6) = 36 - 30 + 1$ $t(6) = 7$

STEP 5

Substitute $g(x) = 4x - 5$ into $t(x)$ :
$t(g(x)) = ((4x - 5)^2) - 5(4x - 5) + 1$

STEP 6

Expand $(4x - 5)^2$ :
$(4x - 5)^2 = (4x)^2 - 2(4x)(5) + 5^2$ $= 16x^2 - 40x + 25$

STEP 7

Substitute the expanded form back into the expression for $t(g(x))$ :
$t(g(x)) = (16x^2 - 40x + 25) - 5(4x - 5) + 1$

STEP 8

Simplify the expression:
$t(g(x)) = 16x^2 - 40x + 25 - (20x - 25) + 1$

STEP 9

Combine like terms:
$t(g(x)) = 16x^2 - 40x + 25 - 20x + 25 + 1$ $t(g(x)) = 16x^2 - 60x + 51$

STEP 10

Verify that the expression matches $16x^2 - 60x + 51$ .
The expression matches, confirming the result.
The value of $t(6)$ is $\boxed{7}$ , and it is shown that $t(g(x)) = 16x^2 - 60x + 51$ .

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QuestionThe functions $g$ and $t$ are defined for $x \in \mathbb{R}$ as follows: $\begin{array}{l} g: x \rightarrow 4x - 5 \\ t: x \rightarrow x^2 - 5x + 1 \end{array}$ (a) Find $t(6)$
(b) Show that $t(g(x)) = 16x^2 - 60x + 51$

Studdy Solution

STEP 1

1. We are given two functions: $g(x) = 4x - 5$ and $t(x) = x^2 - 5x + 1$ .
2. We need to evaluate $t(6)$ .
3. We need to show that $t(g(x)) = 16x^2 - 60x + 51$ .

STEP 2

1. Evaluate $t(6)$ .
2. Substitute $g(x)$ into $t(x)$ to find $t(g(x))$ .
3. Simplify the expression for $t(g(x))$ .
4. Verify that the simplified expression matches $16x^2 - 60x + 51$ .

STEP 3

Substitute $x = 6$ into $t(x)$ :
$t(6) = 6^2 - 5(6) + 1$

STEP 4

Simplify the expression:
$t(6) = 36 - 30 + 1$ $t(6) = 7$

STEP 5

Substitute $g(x) = 4x - 5$ into $t(x)$ :
$t(g(x)) = ((4x - 5)^2) - 5(4x - 5) + 1$

STEP 6

Expand $(4x - 5)^2$ :
$(4x - 5)^2 = (4x)^2 - 2(4x)(5) + 5^2$ $= 16x^2 - 40x + 25$

STEP 7

Substitute the expanded form back into the expression for $t(g(x))$ :
$t(g(x)) = (16x^2 - 40x + 25) - 5(4x - 5) + 1$

STEP 8

Simplify the expression:
$t(g(x)) = 16x^2 - 40x + 25 - (20x - 25) + 1$

STEP 9

Combine like terms:
$t(g(x)) = 16x^2 - 40x + 25 - 20x + 25 + 1$ $t(g(x)) = 16x^2 - 60x + 51$

STEP 10

Verify that the expression matches $16x^2 - 60x + 51$ .
The expression matches, confirming the result.
The value of $t(6)$ is $\boxed{7}$ , and it is shown that $t(g(x)) = 16x^2 - 60x + 51$ .

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Studdy Solution

STEP 1

1. We are given two functions: g(x)=4x−5 g(x) = 4x - 5 g(x)=4x−5 and t(x)=x2−5x+1 t(x) = x^2 - 5x + 1 t(x)=x2−5x+1.2. We need to evaluate t(6) t(6) t(6).3. We need to show that t(g(x))=16x2−60x+51 t(g(x)) = 16x^2 - 60x + 51 t(g(x))=16x2−60x+51.

STEP 2

1. Evaluate t(6) t(6) t(6).2. Substitute g(x) g(x) g(x) into t(x) t(x) t(x) to find t(g(x)) t(g(x)) t(g(x)).3. Simplify the expression for t(g(x)) t(g(x)) t(g(x)).4. Verify that the simplified expression matches 16x2−60x+51 16x^2 - 60x + 51 16x2−60x+51.

STEP 3

Substitute x=6 x = 6 x=6 into t(x) t(x) t(x):t(6)=62−5(6)+1 t(6) = 6^2 - 5(6) + 1 t(6)=62−5(6)+1

STEP 4

Simplify the expression:t(6)=36−30+1 t(6) = 36 - 30 + 1 t(6)=36−30+1 t(6)=7 t(6) = 7 t(6)=7

STEP 5

Substitute g(x)=4x−5 g(x) = 4x - 5 g(x)=4x−5 into t(x) t(x) t(x):t(g(x))=((4x−5)2)−5(4x−5)+1 t(g(x)) = ((4x - 5)^2) - 5(4x - 5) + 1 t(g(x))=((4x−5)2)−5(4x−5)+1

STEP 6

Expand (4x−5)2 (4x - 5)^2 (4x−5)2:(4x−5)2=(4x)2−2(4x)(5)+52 (4x - 5)^2 = (4x)^2 - 2(4x)(5) + 5^2 (4x−5)2=(4x)2−2(4x)(5)+52 =16x2−40x+25 = 16x^2 - 40x + 25 =16x2−40x+25

STEP 7

Substitute the expanded form back into the expression for t(g(x)) t(g(x)) t(g(x)):t(g(x))=(16x2−40x+25)−5(4x−5)+1 t(g(x)) = (16x^2 - 40x + 25) - 5(4x - 5) + 1 t(g(x))=(16x2−40x+25)−5(4x−5)+1

STEP 8

Simplify the expression:t(g(x))=16x2−40x+25−(20x−25)+1 t(g(x)) = 16x^2 - 40x + 25 - (20x - 25) + 1 t(g(x))=16x2−40x+25−(20x−25)+1

STEP 9

Combine like terms:t(g(x))=16x2−40x+25−20x+25+1 t(g(x)) = 16x^2 - 40x + 25 - 20x + 25 + 1 t(g(x))=16x2−40x+25−20x+25+1 t(g(x))=16x2−60x+51 t(g(x)) = 16x^2 - 60x + 51 t(g(x))=16x2−60x+51

STEP 10

Verify that the expression matches 16x2−60x+51 16x^2 - 60x + 51 16x2−60x+51.The expression matches, confirming the result. The value of t(6) t(6) t(6) is 7 \boxed{7} 7​, and it is shown that t(g(x))=16x2−60x+51 t(g(x)) = 16x^2 - 60x + 51 t(g(x))=16x2−60x+51.

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1. We are given two functions: $g(x) = 4x - 5$ and $t(x) = x^2 - 5x + 1$ .
2. We need to evaluate $t(6)$ .
3. We need to show that $t(g(x)) = 16x^2 - 60x + 51$ .

1. Evaluate $t(6)$ .
2. Substitute $g(x)$ into $t(x)$ to find $t(g(x))$ .
3. Simplify the expression for $t(g(x))$ .
4. Verify that the simplified expression matches $16x^2 - 60x + 51$ .

Substitute $x = 6$ into $t(x)$ :
$t(6) = 6^2 - 5(6) + 1$

Simplify the expression:
$t(6) = 36 - 30 + 1$ $t(6) = 7$

Substitute $g(x) = 4x - 5$ into $t(x)$ :
$t(g(x)) = ((4x - 5)^2) - 5(4x - 5) + 1$

Expand $(4x - 5)^2$ :
$(4x - 5)^2 = (4x)^2 - 2(4x)(5) + 5^2$ $= 16x^2 - 40x + 25$

Substitute the expanded form back into the expression for $t(g(x))$ :
$t(g(x)) = (16x^2 - 40x + 25) - 5(4x - 5) + 1$

Simplify the expression:
$t(g(x)) = 16x^2 - 40x + 25 - (20x - 25) + 1$

Combine like terms:
$t(g(x)) = 16x^2 - 40x + 25 - 20x + 25 + 1$ $t(g(x)) = 16x^2 - 60x + 51$

Verify that the expression matches $16x^2 - 60x + 51$ .
The expression matches, confirming the result.
The value of $t(6)$ is $\boxed{7}$ , and it is shown that $t(g(x)) = 16x^2 - 60x + 51$ .